Theorem ttukeylem3 10616

Description: Lemma for ttukey 10623. (Contributed by Mario Carneiro, 11-May-2015.)

Hypotheses

Ref	Expression
ttukeylem.1	⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))⟶1-1onto(∪ 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 8434	. . 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 487	. 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 2764	. . 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 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝑧 = (𝐺 ↾ 𝐶))
6	5	dmeqd 5894	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = dom (𝐺 ↾ 𝐶))
7	1	tfr1 8433	. . . . . . . . 9 ⊢ 𝐺 Fn On
8		onss 7823	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
9	8	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝐶 ⊆ On)
10		fnssres 6684	. . . . . . . . 9 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
11	7, 9, 10	sylancr 599	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐺 ↾ 𝐶) Fn 𝐶)
12	11	fndmd 6666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
13	6, 12	eqtrd 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = 𝐶)
14	13	unieqd 4883	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ dom 𝑧 = ∪ 𝐶)
15	13, 14	eqeq12d 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∪ dom 𝑧 ↔ 𝐶 = ∪ 𝐶))
16	13	eqeq1d 2765	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∅ ↔ 𝐶 = ∅))
17	5	rneqd 5927	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = ran (𝐺 ↾ 𝐶))
18		df-ima 5668	. . . . . . . 8 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
19	17, 18	eqtr4di 2816	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = (𝐺 “ 𝐶))
20	19	unieqd 4883	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ ran 𝑧 = ∪ (𝐺 “ 𝐶))
21	16, 20	ifbieq2d 4512	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧) = if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)))
22	5, 14	fveq12d 6915	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝑧‘∪ dom 𝑧) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
23	14	fveq2d 6912	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐹‘∪ dom 𝑧) = (𝐹‘∪ 𝐶))
24	23	sneqd 4596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → {(𝐹‘∪ dom 𝑧)} = {(𝐹‘∪ 𝐶)})
25	22, 24	uneq12d 4171	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) = (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
26	25	eleq1d 2849	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴 ↔ (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
27		eqidd 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∅ = ∅)
28	26, 24, 27	ifbieq12d 4514	. . . . . 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 4171	. . . . 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 4514	. . . 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 7826	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
32	31	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ On)
33		sucidg 6464	. . . . . . . . 9 ⊢ (∪ 𝐶 ∈ On → ∪ 𝐶 ∈ suc ∪ 𝐶)
34	32, 33	syl 18	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
35		eloni 6390	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → Ord 𝐶)
36	35	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → Ord 𝐶)
37		orduniorsuc 7865	. . . . . . . . . 10 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
38	36, 37	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
39	38	orcanai 1018	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
40	34, 39	eleqtrrd 2867	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
41	40	fvresd 6928	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
42	41	uneq1d 4169	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) = ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
43	42	eleq1d 2849	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴 ↔ ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
44	43	ifbid 4509	. . . . . 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 4171	. . . . 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 4518	. . . 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 2798	. . 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 6661	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
49	7, 48	ax-mp 5	. . . 4 ⊢ Fun 𝐺
50		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
51		resfunexg 7244	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
52	49, 50, 51	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
53		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
54	53	elexd 3477	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
55		funimaexg 6648	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
56	49, 55	mpan 703	. . . . . 6 ⊢ (𝐶 ∈ On → (𝐺 “ 𝐶) ∈ V)
57	56	uniexd 7783	. . . . 5 ⊢ (𝐶 ∈ On → ∪ (𝐺 “ 𝐶) ∈ V)
58		ifcl 4531	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∪ (𝐺 “ 𝐶) ∈ V) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
59	54, 57, 58	syl2an 608	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
60		fvex 6921	. . . . 5 ⊢ (𝐺‘∪ 𝐶) ∈ V
61		snex 5401	. . . . . 6 ⊢ {(𝐹‘∪ 𝐶)} ∈ V
62		0ex 5264	. . . . . 6 ⊢ ∅ ∈ V
63	61, 62	ifex 4536	. . . . 5 ⊢ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅) ∈ V
64	60, 63	unex 7785	. . . 4 ⊢ ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) ∈ V
65		ifcl 4531	. . . 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 598	. . 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 7024	. 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 2798	1 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺‘𝐶) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Ord word 6379  Oncon0 6380  suc csuc 6382  Fun wfun 6552   Fn wfn 6553  ⟶1-1ontowf1o 6557  ‘cfv 6558  recscrecs 8406  Fincfn 9001  cardccrd 10043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407

This theorem is used by:  ttukeylem4  10617  ttukeylem5  10618  ttukeylem6  10619  ttukeylem7  10620