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Theorem fseqenlem2 8731
 Description: Lemma for fseqen 8733. (Contributed by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fseqenlem.a (𝜑𝐴𝑉)
fseqenlem.b (𝜑𝐵𝐴)
fseqenlem.f (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
fseqenlem.g 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
fseqenlem.k 𝐾 = (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
Assertion
Ref Expression
fseqenlem2 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴))
Distinct variable groups:   𝑦,𝐵   𝑓,𝑛,𝑥,𝐹   𝑦,𝑘,𝐺   𝑓,𝑘,𝑦,𝐴,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥,𝑓,𝑘,𝑛)   𝐹(𝑦,𝑘)   𝐺(𝑥,𝑓,𝑛)   𝐾(𝑥,𝑦,𝑓,𝑘,𝑛)   𝑉(𝑥,𝑦,𝑓,𝑘,𝑛)

Proof of Theorem fseqenlem2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 4460 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘))
2 elmapi 7765 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝑚 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 761 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦:𝑘𝐴)
4 fdm 5964 . . . . . . . . 9 (𝑦:𝑘𝐴 → dom 𝑦 = 𝑘)
53, 4syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 = 𝑘)
6 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑘 ∈ ω)
75, 6eqeltrd 2688 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 ∈ ω)
85fveq2d 6107 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
98fveq1d 6105 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺𝑘)‘𝑦))
10 fseqenlem.a . . . . . . . . . . . 12 (𝜑𝐴𝑉)
11 fseqenlem.b . . . . . . . . . . . 12 (𝜑𝐵𝐴)
12 fseqenlem.f . . . . . . . . . . . 12 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
13 fseqenlem.g . . . . . . . . . . . 12 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
1410, 11, 12, 13fseqenlem1 8730 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
1514adantrr 749 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
16 f1f 6014 . . . . . . . . . 10 ((𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴 → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
1715, 16syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
18 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦 ∈ (𝐴𝑚 𝑘))
1917, 18ffvelrnd 6268 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
209, 19eqeltrd 2688 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
21 opelxpi 5072 . . . . . . 7 ((dom 𝑦 ∈ ω ∧ ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
227, 20, 21syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2322rexlimdvaa 3014 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
241, 23syl5bi 231 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2524imp 444 . . 3 ((𝜑𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
26 fseqenlem.k . . 3 𝐾 = (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
2725, 26fmptd 6292 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴))
28 ffun 5961 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
29 funbrfv2b 6150 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3027, 28, 293syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3130simplbda 652 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3230simprbda 651 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
33 fdm 5964 . . . . . . . . . . . . . . . . 17 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3427, 33syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3534adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3632, 35eleqtrd 2690 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘))
37 dmeq 5246 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3837fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
39 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
4038, 39fveq12d 6109 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
4137, 40opeq12d 4348 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42 opex 4859 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4341, 26, 42fvmpt 6191 . . . . . . . . . . . . . 14 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4436, 43syl 17 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4531, 44eqtr3d 2646 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4645fveq2d 6107 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
47 vex 3176 . . . . . . . . . . . . 13 𝑧 ∈ V
4847dmex 6991 . . . . . . . . . . . 12 dom 𝑧 ∈ V
49 fvex 6113 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
5048, 49op1st 7067 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
5146, 50syl6eq 2660 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
5251fveq2d 6107 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5352cnveqd 5220 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5445fveq2d 6107 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5548, 49op2nd 7068 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5654, 55syl6eq 2660 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5753, 56fveq12d 6109 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
58 eliun 4460 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘))
59 elmapi 7765 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴𝑚 𝑘) → 𝑧:𝑘𝐴)
6059adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧:𝑘𝐴)
61 fdm 5964 . . . . . . . . . . . . . . . . 17 (𝑧:𝑘𝐴 → dom 𝑧 = 𝑘)
6260, 61syl 17 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 = 𝑘)
63 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑘 ∈ ω)
6462, 63eqeltrd 2688 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 ∈ ω)
65 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 𝑘))
6662oveq2d 6565 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (𝐴𝑚 dom 𝑧) = (𝐴𝑚 𝑘))
6765, 66eleqtrrd 2691 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
6864, 67jca 553 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
6968rexlimiva 3010 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7058, 69sylbi 206 . . . . . . . . . . . 12 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7136, 70syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7271simpld 474 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
7310, 11, 12, 13fseqenlem1 8730 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
7472, 73syldan 486 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
75 f1f1orn 6061 . . . . . . . . 9 ((𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴 → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7674, 75syl 17 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7771simprd 478 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
78 f1ocnvfv1 6432 . . . . . . . 8 (((𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧) ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7976, 77, 78syl2anc 691 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
8057, 79eqtr2d 2645 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
8180ex 449 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
8281alrimiv 1842 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
83 mo2icl 3352 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8482, 83syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8584alrimiv 1842 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
86 dff12 6013 . 2 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8727, 85, 86sylanbrc 695 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040  suc csuc 5642  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  2nd c2nd 7058  seq𝜔cseqom 7429   ↑𝑚 cmap 7744 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-ima 5051  df-pred 5597  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-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-map 7746 This theorem is referenced by:  fseqen  8733  pwfseqlem5  9364
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