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Theorem axdc3lem2 9156
Description: Lemma for axdc3 9159. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(𝑥 + 1) ∈ 𝐹(𝑠(𝑥)) on its domain, but with the added constraint that 𝑠(0) = 𝐶. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 9151 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 9151 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
Hypotheses
Ref Expression
axdc3lem2.1 𝐴 ∈ V
axdc3lem2.2 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
axdc3lem2.3 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
Assertion
Ref Expression
axdc3lem2 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Distinct variable groups:   𝐴,𝑔,   𝐴,𝑛,𝑠   𝐶,𝑔,   𝐶,𝑛,𝑠   𝑔,𝐹,   𝑛,𝐹,𝑠   𝑘,𝐺   𝑆,𝑘,𝑠   𝑥,𝑆,𝑦   𝑔,𝑘,   ,𝑠   𝑥,,𝑦   𝑘,𝑛
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑘)   𝐶(𝑥,𝑦,𝑘)   𝑆(𝑔,,𝑛)   𝐹(𝑥,𝑦,𝑘)   𝐺(𝑥,𝑦,𝑔,,𝑛,𝑠)

Proof of Theorem axdc3lem2
Dummy variables 𝑖 𝑗 𝑚 𝑢 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . . . . . . 13 (𝑚 = ∅ → 𝑚 = ∅)
2 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (𝑚) = (‘∅))
32dmeqd 5248 . . . . . . . . . . . . 13 (𝑚 = ∅ → dom (𝑚) = dom (‘∅))
41, 3eleq12d 2682 . . . . . . . . . . . 12 (𝑚 = ∅ → (𝑚 ∈ dom (𝑚) ↔ ∅ ∈ dom (‘∅)))
5 eleq2 2677 . . . . . . . . . . . . 13 (𝑚 = ∅ → (𝑗𝑚𝑗 ∈ ∅))
62sseq2d 3596 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘∅)))
75, 6imbi12d 333 . . . . . . . . . . . 12 (𝑚 = ∅ → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
84, 7anbi12d 743 . . . . . . . . . . 11 (𝑚 = ∅ → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅)))))
9 id 22 . . . . . . . . . . . . 13 (𝑚 = 𝑖𝑚 = 𝑖)
10 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚) = (𝑖))
1110dmeqd 5248 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → dom (𝑚) = dom (𝑖))
129, 11eleq12d 2682 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (𝑚 ∈ dom (𝑚) ↔ 𝑖 ∈ dom (𝑖)))
13 elequ2 1991 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → (𝑗𝑚𝑗𝑖))
1410sseq2d 3596 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑖)))
1513, 14imbi12d 333 . . . . . . . . . . . 12 (𝑚 = 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗𝑖 → (𝑗) ⊆ (𝑖))))
1612, 15anbi12d 743 . . . . . . . . . . 11 (𝑚 = 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖)))))
17 id 22 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖𝑚 = suc 𝑖)
18 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚) = (‘suc 𝑖))
1918dmeqd 5248 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → dom (𝑚) = dom (‘suc 𝑖))
2017, 19eleq12d 2682 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
21 eleq2 2677 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑗𝑚𝑗 ∈ suc 𝑖))
2218sseq2d 3596 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘suc 𝑖)))
2321, 22imbi12d 333 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
2420, 23anbi12d 743 . . . . . . . . . . 11 (𝑚 = suc 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
25 peano1 6977 . . . . . . . . . . . . . . 15 ∅ ∈ ω
26 ffvelrn 6265 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆 ∧ ∅ ∈ ω) → (‘∅) ∈ 𝑆)
2725, 26mpan2 703 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (‘∅) ∈ 𝑆)
28 axdc3lem2.2 . . . . . . . . . . . . . . . . . 18 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
29 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠:suc 𝑛𝐴 → dom 𝑠 = suc 𝑛)
30 nnord 6965 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → Ord 𝑛)
31 0elsuc 6927 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝑛 → ∅ ∈ suc 𝑛)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → ∅ ∈ suc 𝑛)
33 peano2 6978 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
34 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ suc 𝑛))
35 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (dom 𝑠 ∈ ω ↔ suc 𝑛 ∈ ω))
3634, 35anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑠 = suc 𝑛 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω)))
3736biimprcd 239 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω) → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3832, 33, 37syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ω → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3929, 38syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠:suc 𝑛𝐴 → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
40393ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
4140impcom 445 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4241rexlimiva 3010 . . . . . . . . . . . . . . . . . . 19 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4342ss2abi 3637 . . . . . . . . . . . . . . . . . 18 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4428, 43eqsstri 3598 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4544sseli 3564 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ 𝑆 → (‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
46 fvex 6113 . . . . . . . . . . . . . . . . 17 (‘∅) ∈ V
47 dmeq 5246 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (‘∅) → dom 𝑠 = dom (‘∅))
4847eleq2d 2673 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘∅)))
4947eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (dom 𝑠 ∈ ω ↔ dom (‘∅) ∈ ω))
5048, 49anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘∅) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω)))
5146, 50elab 3319 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5245, 51sylib 207 . . . . . . . . . . . . . . 15 ((‘∅) ∈ 𝑆 → (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5352simpld 474 . . . . . . . . . . . . . 14 ((‘∅) ∈ 𝑆 → ∅ ∈ dom (‘∅))
5427, 53syl 17 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ∅ ∈ dom (‘∅))
55 noel 3878 . . . . . . . . . . . . . 14 ¬ 𝑗 ∈ ∅
5655pm2.21i 115 . . . . . . . . . . . . 13 (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))
5754, 56jctir 559 . . . . . . . . . . . 12 (:ω⟶𝑆 → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
5857adantr 480 . . . . . . . . . . 11 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
59 ffvelrn 6265 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆𝑖 ∈ ω) → (𝑖) ∈ 𝑆)
6059ancoms 468 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ :ω⟶𝑆) → (𝑖) ∈ 𝑆)
6160adantrr 749 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑖) ∈ 𝑆)
62 suceq 5707 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → suc 𝑘 = suc 𝑖)
6362fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (‘suc 𝑘) = (‘suc 𝑖))
64 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (𝑘) = (𝑖))
6564fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (𝐺‘(𝑘)) = (𝐺‘(𝑖)))
6663, 65eleq12d 2682 . . . . . . . . . . . . . . 15 (𝑘 = 𝑖 → ((‘suc 𝑘) ∈ (𝐺‘(𝑘)) ↔ (‘suc 𝑖) ∈ (𝐺‘(𝑖))))
6766rspcva 3280 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6867adantrl 748 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6944sseli 3564 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → (𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
70 fvex 6113 . . . . . . . . . . . . . . . . . . . . 21 (𝑖) ∈ V
71 dmeq 5246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑖) → dom 𝑠 = dom (𝑖))
7271eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (𝑖)))
7371eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (dom 𝑠 ∈ ω ↔ dom (𝑖) ∈ ω))
7472, 73anbi12d 743 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑖) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω)))
7570, 74elab 3319 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7669, 75sylib 207 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7776simprd 478 . . . . . . . . . . . . . . . . . 18 ((𝑖) ∈ 𝑆 → dom (𝑖) ∈ ω)
78 nnord 6965 . . . . . . . . . . . . . . . . . 18 (dom (𝑖) ∈ ω → Ord dom (𝑖))
79 ordsucelsuc 6914 . . . . . . . . . . . . . . . . . 18 (Ord dom (𝑖) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
8077, 78, 793syl 18 . . . . . . . . . . . . . . . . 17 ((𝑖) ∈ 𝑆 → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
8180adantr 480 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
82 dmeq 5246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑖) → dom 𝑥 = dom (𝑖))
83 suceq 5707 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑥 = dom (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8584eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → (dom 𝑦 = suc dom 𝑥 ↔ dom 𝑦 = suc dom (𝑖)))
8682reseq2d 5317 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → (𝑦 ↾ dom 𝑥) = (𝑦 ↾ dom (𝑖)))
87 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → 𝑥 = (𝑖))
8886, 87eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → ((𝑦 ↾ dom 𝑥) = 𝑥 ↔ (𝑦 ↾ dom (𝑖)) = (𝑖)))
8985, 88anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑖) → ((dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥) ↔ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))))
9089rabbidv 3164 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑖) → {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)} = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
91 axdc3lem2.3 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
92 axdc3lem2.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐴 ∈ V
9392, 28axdc3lem 9155 . . . . . . . . . . . . . . . . . . . . . . 23 𝑆 ∈ V
9493rabex 4740 . . . . . . . . . . . . . . . . . . . . . 22 {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ∈ V
9590, 91, 94fvmpt 6191 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖) ∈ 𝑆 → (𝐺‘(𝑖)) = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
9695eleq2d 2673 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ (‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))}))
97 dmeq 5246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → dom 𝑦 = dom (‘suc 𝑖))
9897eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → (dom 𝑦 = suc dom (𝑖) ↔ dom (‘suc 𝑖) = suc dom (𝑖)))
99 reseq1 5311 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → (𝑦 ↾ dom (𝑖)) = ((‘suc 𝑖) ↾ dom (𝑖)))
10099eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → ((𝑦 ↾ dom (𝑖)) = (𝑖) ↔ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
10198, 100anbi12d 743 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (‘suc 𝑖) → ((dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖)) ↔ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
102101elrab 3331 . . . . . . . . . . . . . . . . . . . 20 ((‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
10396, 102syl6bb 275 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))))
104103simplbda 652 . . . . . . . . . . . . . . . . . 18 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
105104simpld 474 . . . . . . . . . . . . . . . . 17 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → dom (‘suc 𝑖) = suc dom (𝑖))
106105eleq2d 2673 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
10781, 106bitr4d 270 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
108107biimpd 218 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) → suc 𝑖 ∈ dom (‘suc 𝑖)))
109104simprd 478 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))
110 resss 5342 . . . . . . . . . . . . . . . 16 ((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖)
111 sseq1 3589 . . . . . . . . . . . . . . . 16 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
112110, 111mpbii 222 . . . . . . . . . . . . . . 15 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (𝑖) ⊆ (‘suc 𝑖))
113 elsuci 5708 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ suc 𝑖 → (𝑗𝑖𝑗 = 𝑖))
114 pm2.27 41 . . . . . . . . . . . . . . . . . . 19 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗) ⊆ (𝑖)))
115 sstr2 3575 . . . . . . . . . . . . . . . . . . 19 ((𝑗) ⊆ (𝑖) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
116114, 115syl6 34 . . . . . . . . . . . . . . . . . 18 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
117 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑖 → (𝑗) = (𝑖))
118117sseq1d 3595 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → ((𝑗) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
119118biimprd 237 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
120119a1d 25 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
121116, 120jaoi 393 . . . . . . . . . . . . . . . . 17 ((𝑗𝑖𝑗 = 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
122113, 121syl 17 . . . . . . . . . . . . . . . 16 (𝑗 ∈ suc 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
123122com13 86 . . . . . . . . . . . . . . 15 ((𝑖) ⊆ (‘suc 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
124109, 112, 1233syl 18 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
125108, 124anim12d 584 . . . . . . . . . . . . 13 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
12661, 68, 125syl2anc 691 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
127126ex 449 . . . . . . . . . . 11 (𝑖 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))))
1288, 16, 24, 58, 127finds2 6986 . . . . . . . . . 10 (𝑚 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚)))))
129128imp 444 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
130129simprd 478 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑗𝑚 → (𝑗) ⊆ (𝑚)))
131130expcom 450 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
132131ralrimdv 2951 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → ∀𝑗𝑚 (𝑗) ⊆ (𝑚)))
133132ralrimiv 2948 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚))
134 frn 5966 . . . . . . . . . . . 12 (:ω⟶𝑆 → ran 𝑆)
135 ffun 5961 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → Fun 𝑠)
1361353ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
137136rexlimivw 3011 . . . . . . . . . . . . . 14 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
138137ss2abi 3637 . . . . . . . . . . . . 13 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ Fun 𝑠}
13928, 138eqsstri 3598 . . . . . . . . . . . 12 𝑆 ⊆ {𝑠 ∣ Fun 𝑠}
140134, 139syl6ss 3580 . . . . . . . . . . 11 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ Fun 𝑠})
141140sseld 3567 . . . . . . . . . 10 (:ω⟶𝑆 → (𝑢 ∈ ran 𝑢 ∈ {𝑠 ∣ Fun 𝑠}))
142 vex 3176 . . . . . . . . . . 11 𝑢 ∈ V
143 funeq 5823 . . . . . . . . . . 11 (𝑠 = 𝑢 → (Fun 𝑠 ↔ Fun 𝑢))
144142, 143elab 3319 . . . . . . . . . 10 (𝑢 ∈ {𝑠 ∣ Fun 𝑠} ↔ Fun 𝑢)
145141, 144syl6ib 240 . . . . . . . . 9 (:ω⟶𝑆 → (𝑢 ∈ ran → Fun 𝑢))
146145adantr 480 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → Fun 𝑢))
147 ffn 5958 . . . . . . . . 9 (:ω⟶𝑆 Fn ω)
148 fvelrnb 6153 . . . . . . . . . . . . 13 ( Fn ω → (𝑣 ∈ ran ↔ ∃𝑏 ∈ ω (𝑏) = 𝑣))
149 fvelrnb 6153 . . . . . . . . . . . . . . 15 ( Fn ω → (𝑢 ∈ ran ↔ ∃𝑎 ∈ ω (𝑎) = 𝑢))
150 nnord 6965 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ω → Ord 𝑎)
151 nnord 6965 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ ω → Ord 𝑏)
152150, 151anim12i 588 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (Ord 𝑎 ∧ Ord 𝑏))
153 ordtri3or 5672 . . . . . . . . . . . . . . . . . . . . . . 23 ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎𝑏𝑎 = 𝑏𝑏𝑎))
154 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑏 → (𝑚) = (𝑏))
155154sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑏 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑏)))
156155raleqbi1dv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑏 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
157156rspcv 3278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
158 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑎 → (𝑗) = (𝑎))
159158sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑎 → ((𝑗) ⊆ (𝑏) ↔ (𝑎) ⊆ (𝑏)))
160159rspccv 3279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑏 (𝑗) ⊆ (𝑏) → (𝑎𝑏 → (𝑎) ⊆ (𝑏)))
161157, 160syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
162161adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
1631623imp 1249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → (𝑎) ⊆ (𝑏))
164163orcd 406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1651643exp 1256 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
166165com3r 85 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
167 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑏 → (𝑎) = (𝑏))
168 eqimss 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎) = (𝑏) → (𝑎) ⊆ (𝑏))
169168orcd 406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎) = (𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
170167, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1711702a1d 26 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
172 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑎 → (𝑚) = (𝑎))
173172sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑎 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑎)))
174173raleqbi1dv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑎 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
175174rspcv 3278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
176 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑏 → (𝑗) = (𝑏))
177176sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑏 → ((𝑗) ⊆ (𝑎) ↔ (𝑏) ⊆ (𝑎)))
178177rspccv 3279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑎 (𝑗) ⊆ (𝑎) → (𝑏𝑎 → (𝑏) ⊆ (𝑎)))
179175, 178syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
180179adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
1811803imp 1249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → (𝑏) ⊆ (𝑎))
182181olcd 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1831823exp 1256 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
184183com3r 85 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝑎 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
185166, 171, 1843jaoi 1383 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝑏𝑎 = 𝑏𝑏𝑎) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
186153, 185syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord 𝑎 ∧ Ord 𝑏) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
187152, 186mpcom 37 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎))))
188 sseq12 3591 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑎) ⊆ (𝑏) ↔ 𝑢𝑣))
189 sseq12 3591 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏) = 𝑣 ∧ (𝑎) = 𝑢) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
190189ancoms 468 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
191188, 190orbi12d 742 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) ↔ (𝑢𝑣𝑣𝑢)))
192191biimpcd 238 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢)))
193187, 192syl6 34 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢))))
194193com23 84 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
195194exp4b 630 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ω → (𝑏 ∈ ω → ((𝑎) = 𝑢 → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
196195com23 84 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ω → ((𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
197196rexlimiv 3009 . . . . . . . . . . . . . . . 16 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
198197rexlimdv 3012 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
199149, 198syl6bi 242 . . . . . . . . . . . . . 14 ( Fn ω → (𝑢 ∈ ran → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
200199com23 84 . . . . . . . . . . . . 13 ( Fn ω → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
201148, 200sylbid 229 . . . . . . . . . . . 12 ( Fn ω → (𝑣 ∈ ran → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
202201com24 93 . . . . . . . . . . 11 ( Fn ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢)))))
203202imp 444 . . . . . . . . . 10 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢))))
204203ralrimdv 2951 . . . . . . . . 9 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
205147, 204sylan 487 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
206146, 205jcad 554 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢))))
207206ralrimiv 2948 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → ∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
208 fununi 5878 . . . . . 6 (∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)) → Fun ran )
209207, 208syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → Fun ran )
210133, 209syldan 486 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → Fun ran )
211 vex 3176 . . . . . . . . 9 𝑚 ∈ V
212211eldm2 5244 . . . . . . . 8 (𝑚 ∈ dom ran ↔ ∃𝑢𝑚, 𝑢⟩ ∈ ran )
213 eluni2 4376 . . . . . . . . . 10 (⟨𝑚, 𝑢⟩ ∈ ran ↔ ∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣)
214211, 142opeldm 5250 . . . . . . . . . . . . . . 15 (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣)
215214a1i 11 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣))
216134, 44syl6ss 3580 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
217 ssel 3562 . . . . . . . . . . . . . . . 16 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran 𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
218 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
219 dmeq 5246 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑣 → dom 𝑠 = dom 𝑣)
220219eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom 𝑣))
221219eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (dom 𝑠 ∈ ω ↔ dom 𝑣 ∈ ω))
222220, 221anbi12d 743 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑣 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
223218, 222elab 3319 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω))
224223simprbi 479 . . . . . . . . . . . . . . . 16 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → dom 𝑣 ∈ ω)
225217, 224syl6 34 . . . . . . . . . . . . . . 15 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
226216, 225syl 17 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
227215, 226anim12d 584 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → (𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
228 elnn 6967 . . . . . . . . . . . . 13 ((𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω) → 𝑚 ∈ ω)
229227, 228syl6 34 . . . . . . . . . . . 12 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → 𝑚 ∈ ω))
230229expcomd 453 . . . . . . . . . . 11 (:ω⟶𝑆 → (𝑣 ∈ ran → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω)))
231230rexlimdv 3012 . . . . . . . . . 10 (:ω⟶𝑆 → (∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω))
232213, 231syl5bi 231 . . . . . . . . 9 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
233232exlimdv 1848 . . . . . . . 8 (:ω⟶𝑆 → (∃𝑢𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
234212, 233syl5bi 231 . . . . . . 7 (:ω⟶𝑆 → (𝑚 ∈ dom ran 𝑚 ∈ ω))
235234adantr 480 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
236 id 22 . . . . . . . . . . 11 (𝑚 ∈ ω → 𝑚 ∈ ω)
237 fnfvelrn 6264 . . . . . . . . . . 11 (( Fn ω ∧ 𝑚 ∈ ω) → (𝑚) ∈ ran )
238147, 236, 237syl2anr 494 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ :ω⟶𝑆) → (𝑚) ∈ ran )
239238adantrr 749 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚) ∈ ran )
240129simpld 474 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom (𝑚))
241 dmeq 5246 . . . . . . . . . 10 (𝑢 = (𝑚) → dom 𝑢 = dom (𝑚))
242241eliuni 4462 . . . . . . . . 9 (((𝑚) ∈ ran 𝑚 ∈ dom (𝑚)) → 𝑚 𝑢 ∈ ran dom 𝑢)
243239, 240, 242syl2anc 691 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 𝑢 ∈ ran dom 𝑢)
244 dmuni 5256 . . . . . . . 8 dom ran = 𝑢 ∈ ran dom 𝑢
245243, 244syl6eleqr 2699 . . . . . . 7 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom ran )
246245expcom 450 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom ran ))
247235, 246impbid 201 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
248247eqrdv 2608 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → dom ran = ω)
249 rnuni 5463 . . . . . 6 ran ran = 𝑠 ∈ ran ran 𝑠
250 frn 5966 . . . . . . . . . . . . . 14 (𝑠:suc 𝑛𝐴 → ran 𝑠𝐴)
2512503ad2ant1 1075 . . . . . . . . . . . . 13 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
252251rexlimivw 3011 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
253252ss2abi 3637 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ ran 𝑠𝐴}
25428, 253eqsstri 3598 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ ran 𝑠𝐴}
255134, 254syl6ss 3580 . . . . . . . . 9 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ ran 𝑠𝐴})
256 ssel 3562 . . . . . . . . . 10 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran 𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴}))
257 abid 2598 . . . . . . . . . 10 (𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴} ↔ ran 𝑠𝐴)
258256, 257syl6ib 240 . . . . . . . . 9 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran → ran 𝑠𝐴))
259255, 258syl 17 . . . . . . . 8 (:ω⟶𝑆 → (𝑠 ∈ ran → ran 𝑠𝐴))
260259ralrimiv 2948 . . . . . . 7 (:ω⟶𝑆 → ∀𝑠 ∈ ran ran 𝑠𝐴)
261 iunss 4497 . . . . . . 7 ( 𝑠 ∈ ran ran 𝑠𝐴 ↔ ∀𝑠 ∈ ran ran 𝑠𝐴)
262260, 261sylibr 223 . . . . . 6 (:ω⟶𝑆 𝑠 ∈ ran ran 𝑠𝐴)
263249, 262syl5eqss 3612 . . . . 5 (:ω⟶𝑆 → ran ran 𝐴)
264263adantr 480 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran ran 𝐴)
265 df-fn 5807 . . . . 5 ( ran Fn ω ↔ (Fun ran ∧ dom ran = ω))
266 df-f 5808 . . . . . 6 ( ran :ω⟶𝐴 ↔ ( ran Fn ω ∧ ran ran 𝐴))
267266biimpri 217 . . . . 5 (( ran Fn ω ∧ ran ran 𝐴) → ran :ω⟶𝐴)
268265, 267sylanbr 489 . . . 4 (((Fun ran ∧ dom ran = ω) ∧ ran ran 𝐴) → ran :ω⟶𝐴)
269210, 248, 264, 268syl21anc 1317 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran :ω⟶𝐴)
270 fnfvelrn 6264 . . . . . . . 8 (( Fn ω ∧ ∅ ∈ ω) → (‘∅) ∈ ran )
271147, 25, 270sylancl 693 . . . . . . 7 (:ω⟶𝑆 → (‘∅) ∈ ran )
272271adantr 480 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ∈ ran )
273 elssuni 4403 . . . . . 6 ((‘∅) ∈ ran → (‘∅) ⊆ ran )
274272, 273syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ⊆ ran )
27554adantr 480 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∅ ∈ dom (‘∅))
276 funssfv 6119 . . . . 5 ((Fun ran ∧ (‘∅) ⊆ ran ∧ ∅ ∈ dom (‘∅)) → ( ran ‘∅) = ((‘∅)‘∅))
277210, 274, 275, 276syl3anc 1318 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = ((‘∅)‘∅))
278 simp2 1055 . . . . . . . . . . 11 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
279278rexlimivw 3011 . . . . . . . . . 10 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
280279ss2abi 3637 . . . . . . . . 9 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
28128, 280eqsstri 3598 . . . . . . . 8 𝑆 ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
282134, 281syl6ss 3580 . . . . . . 7 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶})
283 ssel 3562 . . . . . . . 8 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → (‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶}))
284 fveq1 6102 . . . . . . . . . 10 (𝑠 = (‘∅) → (𝑠‘∅) = ((‘∅)‘∅))
285284eqeq1d 2612 . . . . . . . . 9 (𝑠 = (‘∅) → ((𝑠‘∅) = 𝐶 ↔ ((‘∅)‘∅) = 𝐶))
28646, 285elab 3319 . . . . . . . 8 ((‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶} ↔ ((‘∅)‘∅) = 𝐶)
287283, 286syl6ib 240 . . . . . . 7 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
288282, 287syl 17 . . . . . 6 (:ω⟶𝑆 → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
289288adantr 480 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
290272, 289mpd 15 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅)‘∅) = 𝐶)
291277, 290eqtrd 2644 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = 𝐶)
292 nfv 1830 . . . . 5 𝑘 :ω⟶𝑆
293 nfra1 2925 . . . . 5 𝑘𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))
294292, 293nfan 1816 . . . 4 𝑘(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))
295134ad2antrr 758 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ran 𝑆)
296 peano2 6978 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
297 fnfvelrn 6264 . . . . . . . . 9 (( Fn ω ∧ suc 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
298147, 296, 297syl2an 493 . . . . . . . 8 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
299298adantlr 747 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
300240expcom 450 . . . . . . . . 9 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom (𝑚)))
301300ralrimiv 2948 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚))
302 id 22 . . . . . . . . . . 11 (𝑚 = suc 𝑘𝑚 = suc 𝑘)
303 fveq2 6103 . . . . . . . . . . . 12 (𝑚 = suc 𝑘 → (𝑚) = (‘suc 𝑘))
304303dmeqd 5248 . . . . . . . . . . 11 (𝑚 = suc 𝑘 → dom (𝑚) = dom (‘suc 𝑘))
305302, 304eleq12d 2682 . . . . . . . . . 10 (𝑚 = suc 𝑘 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
306305rspcv 3278 . . . . . . . . 9 (suc 𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
307296, 306syl 17 . . . . . . . 8 (𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
308301, 307mpan9 485 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ dom (‘suc 𝑘))
309 eleq2 2677 . . . . . . . . . . . . . . . . . . . . 21 (dom 𝑠 = suc 𝑛 → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ suc 𝑛))
310309biimpa 500 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑠 = suc 𝑛 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
31129, 310sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
312 ordsucelsuc 6914 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑛 → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
31330, 312syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ω → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
314313biimprd 237 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛𝑘𝑛))
315 rsp 2913 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑘𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
316314, 315syl9r 76 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
317316com13 86 . . . . . . . . . . . . . . . . . . 19 (suc 𝑘 ∈ suc 𝑛 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
318311, 317syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
319318ex 449 . . . . . . . . . . . . . . . . 17 (𝑠:suc 𝑛𝐴 → (suc 𝑘 ∈ dom 𝑠 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
320319com24 93 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
321320imp 444 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
3223213adant2 1073 . . . . . . . . . . . . . 14 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
323322impcom 445 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
324323rexlimiva 3010 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
325324ss2abi 3637 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
32628, 325eqsstri 3598 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
327 sstr 3576 . . . . . . . . . 10 ((ran 𝑆𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}) → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
328326, 327mpan2 703 . . . . . . . . 9 (ran 𝑆 → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
329328sseld 3567 . . . . . . . 8 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}))
330 fvex 6113 . . . . . . . . 9 (‘suc 𝑘) ∈ V
331 dmeq 5246 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → dom 𝑠 = dom (‘suc 𝑘))
332331eleq2d 2673 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
333 fveq1 6102 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝑠‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
334 fveq1 6102 . . . . . . . . . . . 12 (𝑠 = (‘suc 𝑘) → (𝑠𝑘) = ((‘suc 𝑘)‘𝑘))
335334fveq2d 6107 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝐹‘(𝑠𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
336333, 335eleq12d 2682 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
337332, 336imbi12d 333 . . . . . . . . 9 (𝑠 = (‘suc 𝑘) → ((suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
338330, 337elab 3319 . . . . . . . 8 ((‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
339329, 338syl6ib 240 . . . . . . 7 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
340295, 299, 308, 339syl3c 64 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))
341210adantr 480 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → Fun ran )
342 elssuni 4403 . . . . . . . . . 10 ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ⊆ ran )
343298, 342syl 17 . . . . . . . . 9 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
344343adantlr 747 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
345 funssfv 6119 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
346341, 344, 308, 345syl3anc 1318 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
347216sseld 3567 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
348331eleq2d 2673 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘suc 𝑘)))
349331eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (dom 𝑠 ∈ ω ↔ dom (‘suc 𝑘) ∈ ω))
350348, 349anbi12d 743 . . . . . . . . . . . . . . . 16 (𝑠 = (‘suc 𝑘) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
351330, 350elab 3319 . . . . . . . . . . . . . . 15 ((‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
352347, 351syl6ib 240 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
353352adantr 480 . . . . . . . . . . . . 13 ((:ω⟶𝑆𝑘 ∈ ω) → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
354298, 353mpd 15 . . . . . . . . . . . 12 ((:ω⟶𝑆𝑘 ∈ ω) → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
355354simprd 478 . . . . . . . . . . 11 ((:ω⟶𝑆𝑘 ∈ ω) → dom (‘suc 𝑘) ∈ ω)
356 nnord 6965 . . . . . . . . . . 11 (dom (‘suc 𝑘) ∈ ω → Ord dom (‘suc 𝑘))
357 ordtr 5654 . . . . . . . . . . 11 (Ord dom (‘suc 𝑘) → Tr dom (‘suc 𝑘))
358 trsuc 5727 . . . . . . . . . . . 12 ((Tr dom (‘suc 𝑘) ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → 𝑘 ∈ dom (‘suc 𝑘))
359358ex 449 . . . . . . . . . . 11 (Tr dom (‘suc 𝑘) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
360355, 356, 357, 3594syl 19 . . . . . . . . . 10 ((:ω⟶𝑆𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
361360adantlr 747 . . . . . . . . 9 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
362308, 361mpd 15 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom (‘suc 𝑘))
363 funssfv 6119 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran 𝑘 ∈ dom (‘suc 𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
364341, 344, 362, 363syl3anc 1318 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
365 simpl 472 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
366 simpr 476 . . . . . . . . 9 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
367366fveq2d 6107 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (𝐹‘( ran 𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
368365, 367eleq12d 2682 . . . . . . 7 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
369346, 364, 368syl2anc 691 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
370340, 369mpbird 246 . . . . 5 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
371370ex 449 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑘 ∈ ω → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
372294, 371ralrimi 2940 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
373 vex 3176 . . . . . 6 ∈ V
374373rnex 6992 . . . . 5 ran ∈ V
375374uniex 6851 . . . 4 ran ∈ V
376 feq1 5939 . . . . 5 (𝑔 = ran → (𝑔:ω⟶𝐴 ran :ω⟶𝐴))
377 fveq1 6102 . . . . . 6 (𝑔 = ran → (𝑔‘∅) = ( ran ‘∅))
378377eqeq1d 2612 . . . . 5 (𝑔 = ran → ((𝑔‘∅) = 𝐶 ↔ ( ran ‘∅) = 𝐶))
379 fveq1 6102 . . . . . . 7 (𝑔 = ran → (𝑔‘suc 𝑘) = ( ran ‘suc 𝑘))
380 fveq1 6102 . . . . . . . 8 (𝑔 = ran → (𝑔𝑘) = ( ran 𝑘))
381380fveq2d 6107 . . . . . . 7 (𝑔 = ran → (𝐹‘(𝑔𝑘)) = (𝐹‘( ran 𝑘)))
382379, 381eleq12d 2682 . . . . . 6 (𝑔 = ran → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
383382ralbidv 2969 . . . . 5 (𝑔 = ran → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
384376, 378, 3833anbi123d 1391 . . . 4 (𝑔 = ran → ((𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))) ↔ ( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))))
385375, 384spcev 3273 . . 3 (( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
386269, 291, 372, 385syl3anc 1318 . 2 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
387386exlimiv 1845 1 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  w3o 1030  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  c0 3874  cop 4131   cuni 4372   ciun 4455  cmpt 4643  Tr wtr 4680  dom cdm 5038  ran crn 5039  cres 5040  Ord word 5639  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  wf 5800  cfv 5804  ωcom 6957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-dc 9151
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-om 6958  df-1o 7447
This theorem is referenced by:  axdc3lem4  9158
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