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Theorem aomclem6 36647
Description: Lemma for dfac11 36650. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem6.b 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
aomclem6.c 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem6.d 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem6.e 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
aomclem6.f 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem6.g 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem6.h 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
aomclem6.a (𝜑𝐴 ∈ On)
aomclem6.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem6 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Distinct variable groups:   𝑦,𝑧,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏,𝑐,𝑑,𝑧   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑   𝐴,𝑎,𝑏,𝑐,𝑑,𝑧   𝐻,𝑎,𝑏,𝑐,𝑑,𝑧   𝐺,𝑑
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐)   𝐻(𝑦)

Proof of Theorem aomclem6
StepHypRef Expression
1 ssid 3587 . 2 𝐴𝐴
2 aomclem6.a . . . 4 (𝜑𝐴 ∈ On)
32adantr 480 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ On)
4 sseq1 3589 . . . . . 6 (𝑐 = 𝑑 → (𝑐𝐴𝑑𝐴))
54anbi2d 736 . . . . 5 (𝑐 = 𝑑 → ((𝜑𝑐𝐴) ↔ (𝜑𝑑𝐴)))
6 fveq2 6103 . . . . . 6 (𝑐 = 𝑑 → (𝐻𝑐) = (𝐻𝑑))
7 fveq2 6103 . . . . . 6 (𝑐 = 𝑑 → (𝑅1𝑐) = (𝑅1𝑑))
86, 7weeq12d 36628 . . . . 5 (𝑐 = 𝑑 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑑) We (𝑅1𝑑)))
95, 8imbi12d 333 . . . 4 (𝑐 = 𝑑 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))))
10 sseq1 3589 . . . . . 6 (𝑐 = 𝐴 → (𝑐𝐴𝐴𝐴))
1110anbi2d 736 . . . . 5 (𝑐 = 𝐴 → ((𝜑𝑐𝐴) ↔ (𝜑𝐴𝐴)))
12 fveq2 6103 . . . . . 6 (𝑐 = 𝐴 → (𝐻𝑐) = (𝐻𝐴))
13 fveq2 6103 . . . . . 6 (𝑐 = 𝐴 → (𝑅1𝑐) = (𝑅1𝐴))
1412, 13weeq12d 36628 . . . . 5 (𝑐 = 𝐴 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝐴) We (𝑅1𝐴)))
1511, 14imbi12d 333 . . . 4 (𝑐 = 𝐴 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))))
16 aomclem6.b . . . . . . . . . . . . . 14 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
17 aomclem6.c . . . . . . . . . . . . . 14 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
18 aomclem6.d . . . . . . . . . . . . . 14 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
19 aomclem6.e . . . . . . . . . . . . . 14 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
20 aomclem6.f . . . . . . . . . . . . . 14 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
21 aomclem6.g . . . . . . . . . . . . . 14 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
22 dmeq 5246 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐻𝑐) → dom 𝑧 = dom (𝐻𝑐))
2322adantl 481 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = dom (𝐻𝑐))
24 simpl1 1057 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐 ∈ On)
25 onss 6882 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ On → 𝑐 ⊆ On)
26 aomclem6.h . . . . . . . . . . . . . . . . . . 19 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
2726tfr1 7380 . . . . . . . . . . . . . . . . . 18 𝐻 Fn On
28 fnssres 5918 . . . . . . . . . . . . . . . . . 18 ((𝐻 Fn On ∧ 𝑐 ⊆ On) → (𝐻𝑐) Fn 𝑐)
2927, 28mpan 702 . . . . . . . . . . . . . . . . 17 (𝑐 ⊆ On → (𝐻𝑐) Fn 𝑐)
30 fndm 5904 . . . . . . . . . . . . . . . . 17 ((𝐻𝑐) Fn 𝑐 → dom (𝐻𝑐) = 𝑐)
3124, 25, 29, 304syl 19 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom (𝐻𝑐) = 𝑐)
3223, 31eqtrd 2644 . . . . . . . . . . . . . . 15 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = 𝑐)
3332, 24eqeltrd 2688 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 ∈ On)
3432eleq2d 2673 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎𝑐))
3534biimpa 500 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝑐)
36 simpll2 1094 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)))
37 simpl3l 1109 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝜑)
3837adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝜑)
39 onelss 5683 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑧 ∈ On → (𝑎 ∈ dom 𝑧𝑎 ⊆ dom 𝑧))
4033, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎 ⊆ dom 𝑧))
4140imp 444 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎 ⊆ dom 𝑧)
42 simpl3r 1110 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐𝐴)
4332, 42eqsstrd 3602 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧𝐴)
4443adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → dom 𝑧𝐴)
4541, 44sstrd 3578 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝐴)
46 sseq1 3589 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑑𝐴𝑎𝐴))
4746anbi2d 736 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝜑𝑑𝐴) ↔ (𝜑𝑎𝐴)))
48 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝐻𝑑) = (𝐻𝑎))
49 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑅1𝑑) = (𝑅1𝑎))
5048, 49weeq12d 36628 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝐻𝑑) We (𝑅1𝑑) ↔ (𝐻𝑎) We (𝑅1𝑎)))
5147, 50imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑎 → (((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ↔ ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎))))
5251rspcva 3280 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) → ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎)))
5352imp 444 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) ∧ (𝜑𝑎𝐴)) → (𝐻𝑎) We (𝑅1𝑎))
5435, 36, 38, 45, 53syl22anc 1319 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝐻𝑎) We (𝑅1𝑎))
55 fveq1 6102 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐻𝑐) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
5655ad2antlr 759 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
57 fvres 6117 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑐 → ((𝐻𝑐)‘𝑎) = (𝐻𝑎))
5835, 57syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ((𝐻𝑐)‘𝑎) = (𝐻𝑎))
5956, 58eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = (𝐻𝑎))
60 weeq1 5026 . . . . . . . . . . . . . . . . 17 ((𝑧𝑎) = (𝐻𝑎) → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝐻𝑎) We (𝑅1𝑎)))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝐻𝑎) We (𝑅1𝑎)))
6254, 61mpbird 246 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) We (𝑅1𝑎))
6362ralrimiva 2949 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
6437, 2syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐴 ∈ On)
65 aomclem6.y . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
6637, 65syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
6716, 17, 18, 19, 20, 21, 33, 63, 64, 43, 66aomclem5 36646 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1‘dom 𝑧))
6832fveq2d 6107 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑅1‘dom 𝑧) = (𝑅1𝑐))
69 weeq2 5027 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) = (𝑅1𝑐) → (𝐺 We (𝑅1‘dom 𝑧) ↔ 𝐺 We (𝑅1𝑐)))
7068, 69syl 17 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝐺 We (𝑅1‘dom 𝑧) ↔ 𝐺 We (𝑅1𝑐)))
7167, 70mpbid 221 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1𝑐))
7271ex 449 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
7372alrimiv 1842 . . . . . . . . . 10 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
74 nfv 1830 . . . . . . . . . . 11 𝑑(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐))
75 nfv 1830 . . . . . . . . . . . 12 𝑧 𝑑 = (𝐻𝑐)
76 nfsbc1v 3422 . . . . . . . . . . . 12 𝑧[𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
7775, 76nfim 1813 . . . . . . . . . . 11 𝑧(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
78 eqeq1 2614 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝑧 = (𝐻𝑐) ↔ 𝑑 = (𝐻𝑐)))
79 sbceq1a 3413 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝐺 We (𝑅1𝑐) ↔ [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8078, 79imbi12d 333 . . . . . . . . . . 11 (𝑧 = 𝑑 → ((𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ (𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))))
8174, 77, 80cbval 2259 . . . . . . . . . 10 (∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8273, 81sylib 207 . . . . . . . . 9 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
83 nfsbc1v 3422 . . . . . . . . . 10 𝑑[(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
84 fnfun 5902 . . . . . . . . . . . 12 (𝐻 Fn On → Fun 𝐻)
8527, 84ax-mp 5 . . . . . . . . . . 11 Fun 𝐻
86 vex 3176 . . . . . . . . . . 11 𝑐 ∈ V
87 resfunexg 6384 . . . . . . . . . . 11 ((Fun 𝐻𝑐 ∈ V) → (𝐻𝑐) ∈ V)
8885, 86, 87mp2an 704 . . . . . . . . . 10 (𝐻𝑐) ∈ V
89 sbceq1a 3413 . . . . . . . . . 10 (𝑑 = (𝐻𝑐) → ([𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
9083, 88, 89ceqsal 3205 . . . . . . . . 9 (∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
9182, 90sylib 207 . . . . . . . 8 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
92 sbcco 3425 . . . . . . . 8 ([(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
9391, 92sylib 207 . . . . . . 7 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
94 nfcsb1v 3515 . . . . . . . . . 10 𝑧(𝐻𝑐) / 𝑧𝐺
95 nfcv 2751 . . . . . . . . . 10 𝑧(𝑅1𝑐)
9694, 95nfwe 5014 . . . . . . . . 9 𝑧(𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)
97 csbeq1a 3508 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → 𝐺 = (𝐻𝑐) / 𝑧𝐺)
98 weeq1 5026 . . . . . . . . . 10 (𝐺 = (𝐻𝑐) / 𝑧𝐺 → (𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
9997, 98syl 17 . . . . . . . . 9 (𝑧 = (𝐻𝑐) → (𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
10096, 99sbciegf 3434 . . . . . . . 8 ((𝐻𝑐) ∈ V → ([(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
10188, 100ax-mp 5 . . . . . . 7 ([(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
10293, 101sylib 207 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
103 recsval 7387 . . . . . . . . 9 (𝑐 ∈ On → (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)))
10426fveq1i 6104 . . . . . . . . 9 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐)
105 fvex 6113 . . . . . . . . . . . . . . 15 (𝑅1‘dom 𝑧) ∈ V
106105, 105xpex 6860 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)) ∈ V
107106inex2 4728 . . . . . . . . . . . . 13 (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) ∈ V
10821, 107eqeltri 2684 . . . . . . . . . . . 12 𝐺 ∈ V
109108csbex 4721 . . . . . . . . . . 11 (𝐻𝑐) / 𝑧𝐺 ∈ V
110 eqid 2610 . . . . . . . . . . . 12 (𝑧 ∈ V ↦ 𝐺) = (𝑧 ∈ V ↦ 𝐺)
111110fvmpts 6194 . . . . . . . . . . 11 (((𝐻𝑐) ∈ V ∧ (𝐻𝑐) / 𝑧𝐺 ∈ V) → ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺)
11288, 109, 111mp2an 704 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺
11326reseq1i 5313 . . . . . . . . . . 11 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)
114113fveq2i 6106 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
115112, 114eqtr3i 2634 . . . . . . . . 9 (𝐻𝑐) / 𝑧𝐺 = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
116103, 104, 1153eqtr4g 2669 . . . . . . . 8 (𝑐 ∈ On → (𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺)
117 weeq1 5026 . . . . . . . 8 ((𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
118116, 117syl 17 . . . . . . 7 (𝑐 ∈ On → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
1191183ad2ant1 1075 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
120102, 119mpbird 246 . . . . 5 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) We (𝑅1𝑐))
1211203exp 1256 . . . 4 (𝑐 ∈ On → (∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) → ((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐))))
1229, 15, 121tfis3 6949 . . 3 (𝐴 ∈ On → ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴)))
1233, 122mpcom 37 . 2 ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))
1241, 123mpan2 703 1 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  [wsbc 3402  csb 3499  cdif 3537  cin 3539  wss 3540  c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125   cuni 4372   cint 4410   class class class wbr 4583  {copab 4642  cmpt 4643   E cep 4947   We wwe 4996   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  cfv 5804  recscrecs 7354  Fincfn 7841  supcsup 8229  𝑅1cr1 8508  rankcrnk 8509
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-fal 1481  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-rmo 2904  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-int 4411  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-isom 5813  df-riota 6511  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-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-sup 8231  df-r1 8510  df-rank 8511
This theorem is referenced by:  aomclem7  36648
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