Theorem ttukeylem3 10456

Description: Lemma for ttukey 10463. (Contributed by Mario Carneiro, 11-May-2015.)

Hypotheses

Ref	Expression
ttukeylem.1	⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
ttukeylem.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
ttukeylem.3	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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	⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺‘𝐶) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((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

Step	Hyp	Ref	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)}, ∅)))))
2	1	tfr2 8349	. . 3 ⊢ (𝐶 ∈ On → (𝐺‘𝐶) = ((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)}, ∅))))‘(𝐺 ↾ 𝐶)))
3	2	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺‘𝐶) = ((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)}, ∅))))‘(𝐺 ↾ 𝐶)))
4		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 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 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝑧 = (𝐺 ↾ 𝐶))
6	5	dmeqd 5866	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = dom (𝐺 ↾ 𝐶))
7	1	tfr1 8348	. . . . . . . . 9 ⊢ 𝐺 Fn On
8		onss 7724	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
9	8	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝐶 ⊆ On)
10		fnssres 6629	. . . . . . . . 9 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
11	7, 9, 10	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐺 ↾ 𝐶) Fn 𝐶)
12	11	fndmd 6612	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
13	6, 12	eqtrd 2771	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = 𝐶)
14	13	unieqd 4884	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ dom 𝑧 = ∪ 𝐶)
15	13, 14	eqeq12d 2747	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∪ dom 𝑧 ↔ 𝐶 = ∪ 𝐶))
16	13	eqeq1d 2733	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∅ ↔ 𝐶 = ∅))
17	5	rneqd 5898	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = ran (𝐺 ↾ 𝐶))
18		df-ima 5651	. . . . . . . 8 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
19	17, 18	eqtr4di 2789	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = (𝐺 “ 𝐶))
20	19	unieqd 4884	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ ran 𝑧 = ∪ (𝐺 “ 𝐶))
21	16, 20	ifbieq2d 4517	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧) = if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)))
22	5, 14	fveq12d 6854	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝑧‘∪ dom 𝑧) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
23	14	fveq2d 6851	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐹‘∪ dom 𝑧) = (𝐹‘∪ 𝐶))
24	23	sneqd 4603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → {(𝐹‘∪ dom 𝑧)} = {(𝐹‘∪ 𝐶)})
25	22, 24	uneq12d 4129	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) = (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
26	25	eleq1d 2817	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴 ↔ (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
27		eqidd 2732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∅ = ∅)
28	26, 24, 27	ifbieq12d 4519	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅) = if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))
29	22, 28	uneq12d 4129	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅)) = (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)))
30	15, 21, 29	ifbieq12d 4519	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 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, ∪ (𝐺 “ 𝐶)), (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))
31		onuni 7728	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
32	31	ad3antlr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ On)
33		sucidg 6403	. . . . . . . . 9 ⊢ (∪ 𝐶 ∈ On → ∪ 𝐶 ∈ suc ∪ 𝐶)
34	32, 33	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
35		eloni 6332	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → Ord 𝐶)
36	35	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → Ord 𝐶)
37		orduniorsuc 7770	. . . . . . . . . 10 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
39	38	orcanai 1001	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
40	34, 39	eleqtrrd 2835	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
41	40	fvresd 6867	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
42	41	uneq1d 4127	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) = ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
43	42	eleq1d 2817	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴 ↔ ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
44	43	ifbid 4514	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅) = if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))
45	41, 44	uneq12d 4129	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) = ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)))
46	45	ifeq2da 4523	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ if((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))
47	30, 46	eqtrd 2771	. . 3 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 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, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))
48		fnfun 6607	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
49	7, 48	ax-mp 5	. . . 4 ⊢ Fun 𝐺
50		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
51		resfunexg 7170	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
52	49, 50, 51	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
53		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
54	53	elexd 3466	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
55		funimaexg 6592	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
56	49, 55	mpan 688	. . . . . 6 ⊢ (𝐶 ∈ On → (𝐺 “ 𝐶) ∈ V)
57	56	uniexd 7684	. . . . 5 ⊢ (𝐶 ∈ On → ∪ (𝐺 “ 𝐶) ∈ V)
58		ifcl 4536	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∪ (𝐺 “ 𝐶) ∈ V) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
59	54, 57, 58	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
60		fvex 6860	. . . . 5 ⊢ (𝐺‘∪ 𝐶) ∈ V
61		snex 5393	. . . . . 6 ⊢ {(𝐹‘∪ 𝐶)} ∈ V
62		0ex 5269	. . . . . 6 ⊢ ∅ ∈ V
63	61, 62	ifex 4541	. . . . 5 ⊢ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅) ∈ V
64	60, 63	unex 7685	. . . 4 ⊢ ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) ∈ V
65		ifcl 4536	. . . 4 ⊢ ((if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V ∧ ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) ∈ V) → if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))) ∈ V)
66	59, 64, 65	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))) ∈ V)
67	4, 47, 52, 66	fvmptd 6960	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 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)}, ∅))))‘(𝐺 ↾ 𝐶)) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))
68	3, 67	eqtrd 2771	1 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺‘𝐶) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541   ∈ wcel 2106  Vcvv 3446   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  ifcif 4491  𝒫 cpw 4565  {csn 4591  ∪ cuni 4870   ↦ cmpt 5193  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Ord word 6321  Oncon0 6322  suc csuc 6324  Fun wfun 6495   Fn wfn 6496  –1-1-onto→wf1o 6500  ‘cfv 6501  recscrecs 8321  Fincfn 8890  cardccrd 9880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322

This theorem is referenced by:  ttukeylem4  10457  ttukeylem5  10458  ttukeylem6  10459  ttukeylem7  10460