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Theorem ttukeylem3 9922
Description: Lemma for ttukey 9929. (Contributed by Mario Carneiro, 11-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
ttukeylem.4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
Assertion
Ref Expression
ttukeylem3 ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
Distinct variable groups:   𝑥,𝑧,𝐶   𝑥,𝐺,𝑧   𝜑,𝑧   𝑥,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝐹,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem3
StepHypRef Expression
1 ttukeylem.4 . . . 4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
21tfr2 8025 . . 3 (𝐶 ∈ On → (𝐺𝐶) = ((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅))))‘(𝐺𝐶)))
32adantl 482 . 2 ((𝜑𝐶 ∈ On) → (𝐺𝐶) = ((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅))))‘(𝐺𝐶)))
4 eqidd 2827 . . 3 ((𝜑𝐶 ∈ On) → (𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))) = (𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
5 simpr 485 . . . . . . . 8 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → 𝑧 = (𝐺𝐶))
65dmeqd 5773 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → dom 𝑧 = dom (𝐺𝐶))
71tfr1 8024 . . . . . . . . 9 𝐺 Fn On
8 onss 7493 . . . . . . . . . 10 (𝐶 ∈ On → 𝐶 ⊆ On)
98ad2antlr 723 . . . . . . . . 9 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → 𝐶 ⊆ On)
10 fnssres 6467 . . . . . . . . 9 ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺𝐶) Fn 𝐶)
117, 9, 10sylancr 587 . . . . . . . 8 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (𝐺𝐶) Fn 𝐶)
12 fndm 6452 . . . . . . . 8 ((𝐺𝐶) Fn 𝐶 → dom (𝐺𝐶) = 𝐶)
1311, 12syl 17 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → dom (𝐺𝐶) = 𝐶)
146, 13eqtrd 2861 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → dom 𝑧 = 𝐶)
1514unieqd 4847 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → dom 𝑧 = 𝐶)
1614, 15eqeq12d 2842 . . . . 5 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (dom 𝑧 = dom 𝑧𝐶 = 𝐶))
1714eqeq1d 2828 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (dom 𝑧 = ∅ ↔ 𝐶 = ∅))
185rneqd 5807 . . . . . . . 8 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ran 𝑧 = ran (𝐺𝐶))
19 df-ima 5567 . . . . . . . 8 (𝐺𝐶) = ran (𝐺𝐶)
2018, 19syl6eqr 2879 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ran 𝑧 = (𝐺𝐶))
2120unieqd 4847 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ran 𝑧 = (𝐺𝐶))
2217, 21ifbieq2d 4495 . . . . 5 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → if(dom 𝑧 = ∅, 𝐵, ran 𝑧) = if(𝐶 = ∅, 𝐵, (𝐺𝐶)))
235, 15fveq12d 6674 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (𝑧 dom 𝑧) = ((𝐺𝐶)‘ 𝐶))
2415fveq2d 6671 . . . . . . . . . 10 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (𝐹 dom 𝑧) = (𝐹 𝐶))
2524sneqd 4576 . . . . . . . . 9 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → {(𝐹 dom 𝑧)} = {(𝐹 𝐶)})
2623, 25uneq12d 4144 . . . . . . . 8 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) = (((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}))
2726eleq1d 2902 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴 ↔ (((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴))
28 eqidd 2827 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ∅ = ∅)
2927, 25, 28ifbieq12d 4497 . . . . . 6 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅) = if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))
3023, 29uneq12d 4144 . . . . 5 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)) = (((𝐺𝐶)‘ 𝐶) ∪ if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅)))
3116, 22, 30ifbieq12d 4497 . . . 4 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅))) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), (((𝐺𝐶)‘ 𝐶) ∪ if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
32 onuni 7496 . . . . . . . . . 10 (𝐶 ∈ On → 𝐶 ∈ On)
3332ad3antlr 727 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ On)
34 sucidg 6267 . . . . . . . . 9 ( 𝐶 ∈ On → 𝐶 ∈ suc 𝐶)
3533, 34syl 17 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ suc 𝐶)
36 eloni 6199 . . . . . . . . . . 11 (𝐶 ∈ On → Ord 𝐶)
3736ad2antlr 723 . . . . . . . . . 10 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → Ord 𝐶)
38 orduniorsuc 7533 . . . . . . . . . 10 (Ord 𝐶 → (𝐶 = 𝐶𝐶 = suc 𝐶))
3937, 38syl 17 . . . . . . . . 9 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → (𝐶 = 𝐶𝐶 = suc 𝐶))
4039orcanai 998 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 = suc 𝐶)
4135, 40eleqtrrd 2921 . . . . . . 7 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶𝐶)
4241fvresd 6687 . . . . . 6 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
4342uneq1d 4142 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → (((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) = ((𝐺 𝐶) ∪ {(𝐹 𝐶)}))
4443eleq1d 2902 . . . . . . 7 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴 ↔ ((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴))
4544ifbid 4492 . . . . . 6 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅) = if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))
4642, 45uneq12d 4144 . . . . 5 ((((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) ∧ ¬ 𝐶 = 𝐶) → (((𝐺𝐶)‘ 𝐶) ∪ if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅)) = ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅)))
4746ifeq2da 4501 . . . 4 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), (((𝐺𝐶)‘ 𝐶) ∪ if((((𝐺𝐶)‘ 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
4831, 47eqtrd 2861 . . 3 (((𝜑𝐶 ∈ On) ∧ 𝑧 = (𝐺𝐶)) → if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅))) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
49 fnfun 6450 . . . . 5 (𝐺 Fn On → Fun 𝐺)
507, 49ax-mp 5 . . . 4 Fun 𝐺
51 simpr 485 . . . 4 ((𝜑𝐶 ∈ On) → 𝐶 ∈ On)
52 resfunexg 6973 . . . 4 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
5350, 51, 52sylancr 587 . . 3 ((𝜑𝐶 ∈ On) → (𝐺𝐶) ∈ V)
54 ttukeylem.2 . . . . . 6 (𝜑𝐵𝐴)
5554elexd 3520 . . . . 5 (𝜑𝐵 ∈ V)
56 funimaexg 6437 . . . . . . 7 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
5750, 56mpan 686 . . . . . 6 (𝐶 ∈ On → (𝐺𝐶) ∈ V)
58 uniexg 7457 . . . . . 6 ((𝐺𝐶) ∈ V → (𝐺𝐶) ∈ V)
5957, 58syl 17 . . . . 5 (𝐶 ∈ On → (𝐺𝐶) ∈ V)
60 ifcl 4514 . . . . 5 ((𝐵 ∈ V ∧ (𝐺𝐶) ∈ V) → if(𝐶 = ∅, 𝐵, (𝐺𝐶)) ∈ V)
6155, 59, 60syl2an 595 . . . 4 ((𝜑𝐶 ∈ On) → if(𝐶 = ∅, 𝐵, (𝐺𝐶)) ∈ V)
62 fvex 6680 . . . . 5 (𝐺 𝐶) ∈ V
63 snex 5328 . . . . . 6 {(𝐹 𝐶)} ∈ V
64 0ex 5208 . . . . . 6 ∅ ∈ V
6563, 64ifex 4518 . . . . 5 if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅) ∈ V
6662, 65unex 7458 . . . 4 ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅)) ∈ V
67 ifcl 4514 . . . 4 ((if(𝐶 = ∅, 𝐵, (𝐺𝐶)) ∈ V ∧ ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅)) ∈ V) → if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))) ∈ V)
6861, 66, 67sylancl 586 . . 3 ((𝜑𝐶 ∈ On) → if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))) ∈ V)
694, 48, 53, 68fvmptd 6771 . 2 ((𝜑𝐶 ∈ On) → ((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅))))‘(𝐺𝐶)) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
703, 69eqtrd 2861 1 ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  wal 1528   = wceq 1530  wcel 2107  Vcvv 3500  cdif 3937  cun 3938  cin 3939  wss 3940  c0 4295  ifcif 4470  𝒫 cpw 4542  {csn 4564   cuni 4837  cmpt 5143  dom cdm 5554  ran crn 5555  cres 5556  cima 5557  Ord word 6188  Oncon0 6189  suc csuc 6191  Fun wfun 6346   Fn wfn 6347  1-1-ontowf1o 6351  cfv 6352  recscrecs 7998  Fincfn 8498  cardccrd 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-wrecs 7938  df-recs 7999
This theorem is referenced by:  ttukeylem4  9923  ttukeylem5  9924  ttukeylem6  9925  ttukeylem7  9926
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