Theorem ttukeylem3 10465

Description: Lemma for ttukey 10472. (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 8364	. . 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 485	. 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 2762	. . 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 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝑧 = (𝐺 ↾ 𝐶))
6	5	dmeqd 5879	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = dom (𝐺 ↾ 𝐶))
7	1	tfr1 8363	. . . . . . . . 9 ⊢ 𝐺 Fn On
8		onss 7764	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
9	8	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝐶 ⊆ On)
10		fnssres 6640	. . . . . . . . 9 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
11	7, 9, 10	sylancr 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐺 ↾ 𝐶) Fn 𝐶)
12	11	fndmd 6622	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
13	6, 12	eqtrd 2796	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = 𝐶)
14	13	unieqd 4877	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ dom 𝑧 = ∪ 𝐶)
15	13, 14	eqeq12d 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∪ dom 𝑧 ↔ 𝐶 = ∪ 𝐶))
16	13	eqeq1d 2763	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∅ ↔ 𝐶 = ∅))
17	5	rneqd 5912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = ran (𝐺 ↾ 𝐶))
18		df-ima 5658	. . . . . . . 8 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
19	17, 18	eqtr4di 2814	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = (𝐺 “ 𝐶))
20	19	unieqd 4877	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ ran 𝑧 = ∪ (𝐺 “ 𝐶))
21	16, 20	ifbieq2d 4506	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧) = if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)))
22	5, 14	fveq12d 6870	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝑧‘∪ dom 𝑧) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
23	14	fveq2d 6867	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐹‘∪ dom 𝑧) = (𝐹‘∪ 𝐶))
24	23	sneqd 4593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → {(𝐹‘∪ dom 𝑧)} = {(𝐹‘∪ 𝐶)})
25	22, 24	uneq12d 4122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) = (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
26	25	eleq1d 2846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴 ↔ (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
27		eqidd 2762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∅ = ∅)
28	26, 24, 27	ifbieq12d 4508	. . . . . 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 4122	. . . . 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 4508	. . . 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 7767	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
32	31	ad3antlr 741	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ On)
33		sucidg 6425	. . . . . . . . 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 6352	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → Ord 𝐶)
36	35	ad2antlr 737	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → Ord 𝐶)
37		orduniorsuc 7806	. . . . . . . . . 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 1015	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
40	34, 39	eleqtrrd 2864	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
41	40	fvresd 6883	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
42	41	uneq1d 4120	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) = ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
43	42	eleq1d 2846	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴 ↔ ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
44	43	ifbid 4503	. . . . . 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 4122	. . . . 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 4512	. . . 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 2796	. . 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 6617	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
49	7, 48	ax-mp 5	. . . 4 ⊢ Fun 𝐺
50		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
51		resfunexg 7195	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
52	49, 50, 51	sylancr 596	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
53		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
54	53	elexd 3476	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
55		funimaexg 6604	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
56	49, 55	mpan 700	. . . . . 6 ⊢ (𝐶 ∈ On → (𝐺 “ 𝐶) ∈ V)
57	56	uniexd 7721	. . . . 5 ⊢ (𝐶 ∈ On → ∪ (𝐺 “ 𝐶) ∈ V)
58		ifcl 4525	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∪ (𝐺 “ 𝐶) ∈ V) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
59	54, 57, 58	syl2an 605	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
60		fvex 6876	. . . . 5 ⊢ (𝐺‘∪ 𝐶) ∈ V
61		snex 5395	. . . . . 6 ⊢ {(𝐹‘∪ 𝐶)} ∈ V
62		0ex 5256	. . . . . 6 ⊢ ∅ ∈ V
63	61, 62	ifex 4530	. . . . 5 ⊢ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅) ∈ V
64	60, 63	unex 7723	. . . 4 ⊢ ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) ∈ V
65		ifcl 4525	. . . 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 595	. . 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 6979	. 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 2796	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 208   ∧ wa 399   ∨ wo 858  ∀wal 1557   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ifcif 4479  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864   ↦ cmpt 5180  dom cdm 5645  ran crn 5646   ↾ cres 5647   “ cima 5648  Ord word 6341  Oncon0 6342  suc csuc 6344  Fun wfun 6511   Fn wfn 6512  –1-1-onto→wf1o 6516  ‘cfv 6517  recscrecs 8336  Fincfn 8923  cardccrd 9890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337

This theorem is referenced by:  ttukeylem4  10466  ttukeylem5  10467  ttukeylem6  10468  ttukeylem7  10469