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Theorem aomclem6 43071
Description: Lemma for dfac11 43074. 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 4006 . 2 𝐴𝐴
2 aomclem6.a . . . 4 (𝜑𝐴 ∈ On)
32adantr 480 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ On)
4 sseq1 4009 . . . . . 6 (𝑐 = 𝑑 → (𝑐𝐴𝑑𝐴))
54anbi2d 630 . . . . 5 (𝑐 = 𝑑 → ((𝜑𝑐𝐴) ↔ (𝜑𝑑𝐴)))
6 fveq2 6906 . . . . . 6 (𝑐 = 𝑑 → (𝐻𝑐) = (𝐻𝑑))
7 fveq2 6906 . . . . . 6 (𝑐 = 𝑑 → (𝑅1𝑐) = (𝑅1𝑑))
86, 7weeq12d 5674 . . . . 5 (𝑐 = 𝑑 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑑) We (𝑅1𝑑)))
95, 8imbi12d 344 . . . 4 (𝑐 = 𝑑 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))))
10 sseq1 4009 . . . . . 6 (𝑐 = 𝐴 → (𝑐𝐴𝐴𝐴))
1110anbi2d 630 . . . . 5 (𝑐 = 𝐴 → ((𝜑𝑐𝐴) ↔ (𝜑𝐴𝐴)))
12 fveq2 6906 . . . . . 6 (𝑐 = 𝐴 → (𝐻𝑐) = (𝐻𝐴))
13 fveq2 6906 . . . . . 6 (𝑐 = 𝐴 → (𝑅1𝑐) = (𝑅1𝐴))
1412, 13weeq12d 5674 . . . . 5 (𝑐 = 𝐴 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝐴) We (𝑅1𝐴)))
1511, 14imbi12d 344 . . . 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 5914 . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐 ∈ On)
25 onss 7805 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ On → 𝑐 ⊆ On)
26 aomclem6.h . . . . . . . . . . . . . . . . . . 19 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
2726tfr1 8437 . . . . . . . . . . . . . . . . . 18 𝐻 Fn On
28 fnssres 6691 . . . . . . . . . . . . . . . . . 18 ((𝐻 Fn On ∧ 𝑐 ⊆ On) → (𝐻𝑐) Fn 𝑐)
2927, 28mpan 690 . . . . . . . . . . . . . . . . 17 (𝑐 ⊆ On → (𝐻𝑐) Fn 𝑐)
30 fndm 6671 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . 15 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = 𝑐)
3332, 24eqeltrd 2841 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 ∈ On)
3432eleq2d 2827 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎𝑐))
3534biimpa 476 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝑐)
36 simpll2 1214 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)))
37 simpl3l 1229 . . . . . . . . . . . . . . . . . 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 6426 . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎 ⊆ dom 𝑧)
42 simpl3r 1230 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐𝐴)
4332, 42eqsstrd 4018 . . . . . . . . . . . . . . . . . . 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 3994 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝐴)
46 sseq1 4009 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑑𝐴𝑎𝐴))
4746anbi2d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝜑𝑑𝐴) ↔ (𝜑𝑎𝐴)))
48 fveq2 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝐻𝑑) = (𝐻𝑎))
49 fveq2 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑅1𝑑) = (𝑅1𝑎))
5048, 49weeq12d 5674 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝐻𝑑) We (𝑅1𝑑) ↔ (𝐻𝑎) We (𝑅1𝑎)))
5147, 50imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑎 → (((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ↔ ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎))))
5251rspcva 3620 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) → ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎)))
5352imp 406 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) ∧ (𝜑𝑎𝐴)) → (𝐻𝑎) We (𝑅1𝑎))
5435, 36, 38, 45, 53syl22anc 839 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝐻𝑎) We (𝑅1𝑎))
55 fveq1 6905 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐻𝑐) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
5655ad2antlr 727 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
57 fvres 6925 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = (𝐻𝑎))
60 weeq1 5672 . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) We (𝑅1𝑎))
6362ralrimiva 3146 . . . . . . . . . . . . . 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 43070 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1‘dom 𝑧))
6832fveq2d 6910 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑅1‘dom 𝑧) = (𝑅1𝑐))
69 weeq2 5673 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1𝑐))
7271ex 412 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
7372alrimiv 1927 . . . . . . . . . 10 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
74 nfv 1914 . . . . . . . . . . 11 𝑑(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐))
75 nfv 1914 . . . . . . . . . . . 12 𝑧 𝑑 = (𝐻𝑐)
76 nfsbc1v 3808 . . . . . . . . . . . 12 𝑧[𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
7775, 76nfim 1896 . . . . . . . . . . 11 𝑧(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
78 eqeq1 2741 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝑧 = (𝐻𝑐) ↔ 𝑑 = (𝐻𝑐)))
79 sbceq1a 3799 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝐺 We (𝑅1𝑐) ↔ [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8078, 79imbi12d 344 . . . . . . . . . . 11 (𝑧 = 𝑑 → ((𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ (𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))))
8174, 77, 80cbvalv1 2343 . . . . . . . . . 10 (∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8273, 81sylib 218 . . . . . . . . 9 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
83 nfsbc1v 3808 . . . . . . . . . 10 𝑑[(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
84 fnfun 6668 . . . . . . . . . . . 12 (𝐻 Fn On → Fun 𝐻)
8527, 84ax-mp 5 . . . . . . . . . . 11 Fun 𝐻
86 vex 3484 . . . . . . . . . . 11 𝑐 ∈ V
87 resfunexg 7235 . . . . . . . . . . 11 ((Fun 𝐻𝑐 ∈ V) → (𝐻𝑐) ∈ V)
8885, 86, 87mp2an 692 . . . . . . . . . 10 (𝐻𝑐) ∈ V
89 sbceq1a 3799 . . . . . . . . . 10 (𝑑 = (𝐻𝑐) → ([𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
9083, 88, 89ceqsal 3519 . . . . . . . . 9 (∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
9182, 90sylib 218 . . . . . . . 8 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
92 sbccow 3811 . . . . . . . 8 ([(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
9391, 92sylib 218 . . . . . . 7 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
94 nfcsb1v 3923 . . . . . . . . . 10 𝑧(𝐻𝑐) / 𝑧𝐺
95 nfcv 2905 . . . . . . . . . 10 𝑧(𝑅1𝑐)
9694, 95nfwe 5660 . . . . . . . . 9 𝑧(𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)
97 csbeq1a 3913 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → 𝐺 = (𝐻𝑐) / 𝑧𝐺)
98 weeq1 5672 . . . . . . . . . 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 3827 . . . . . . . 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 218 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
103 recsval 8444 . . . . . . . . 9 (𝑐 ∈ On → (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)))
10426fveq1i 6907 . . . . . . . . 9 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐)
105 fvex 6919 . . . . . . . . . . . . . . 15 (𝑅1‘dom 𝑧) ∈ V
106105, 105xpex 7773 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)) ∈ V
107106inex2 5318 . . . . . . . . . . . . 13 (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) ∈ V
10821, 107eqeltri 2837 . . . . . . . . . . . 12 𝐺 ∈ V
109108csbex 5311 . . . . . . . . . . 11 (𝐻𝑐) / 𝑧𝐺 ∈ V
110 eqid 2737 . . . . . . . . . . . 12 (𝑧 ∈ V ↦ 𝐺) = (𝑧 ∈ V ↦ 𝐺)
111110fvmpts 7019 . . . . . . . . . . 11 (((𝐻𝑐) ∈ V ∧ (𝐻𝑐) / 𝑧𝐺 ∈ V) → ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺)
11288, 109, 111mp2an 692 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺
11326reseq1i 5993 . . . . . . . . . . 11 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)
114113fveq2i 6909 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
115112, 114eqtr3i 2767 . . . . . . . . 9 (𝐻𝑐) / 𝑧𝐺 = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
116103, 104, 1153eqtr4g 2802 . . . . . . . 8 (𝑐 ∈ On → (𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺)
117 weeq1 5672 . . . . . . . 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 1134 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
120102, 119mpbird 257 . . . . 5 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) We (𝑅1𝑐))
1211203exp 1120 . . . 4 (𝑐 ∈ On → (∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) → ((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐))))
1229, 15, 121tfis3 7879 . . 3 (𝐴 ∈ On → ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴)))
1233, 122mpcom 38 . 2 ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))
1241, 123mpan2 691 1 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  [wsbc 3788  csb 3899  cdif 3948  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626   cuni 4907   cint 4946   class class class wbr 5143  {copab 5205  cmpt 5225   E cep 5583   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Oncon0 6384  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  cfv 6561  recscrecs 8410  Fincfn 8985  supcsup 9480  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-fin 8989  df-sup 9482  df-r1 9804  df-rank 9805
This theorem is referenced by:  aomclem7  43072
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