Theorem ttukeylem3 9333

Description: Lemma for ttukey 9340. (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 7494	. . 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 2623	. . 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 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝑧 = (𝐺 ↾ 𝐶))
6	5	dmeqd 5326	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = dom (𝐺 ↾ 𝐶))
7	1	tfr1 7493	. . . . . . . . 9 ⊢ 𝐺 Fn On
8		onss 6990	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
9	8	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → 𝐶 ⊆ On)
10		fnssres 6004	. . . . . . . . 9 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
11	7, 9, 10	sylancr 695	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐺 ↾ 𝐶) Fn 𝐶)
12		fndm 5990	. . . . . . . 8 ⊢ ((𝐺 ↾ 𝐶) Fn 𝐶 → dom (𝐺 ↾ 𝐶) = 𝐶)
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
14	6, 13	eqtrd 2656	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → dom 𝑧 = 𝐶)
15	14	unieqd 4446	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ dom 𝑧 = ∪ 𝐶)
16	14, 15	eqeq12d 2637	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∪ dom 𝑧 ↔ 𝐶 = ∪ 𝐶))
17	14	eqeq1d 2624	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (dom 𝑧 = ∅ ↔ 𝐶 = ∅))
18	5	rneqd 5353	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = ran (𝐺 ↾ 𝐶))
19		df-ima 5127	. . . . . . . 8 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
20	18, 19	syl6eqr 2674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ran 𝑧 = (𝐺 “ 𝐶))
21	20	unieqd 4446	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∪ ran 𝑧 = ∪ (𝐺 “ 𝐶))
22	17, 21	ifbieq2d 4111	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧) = if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)))
23	5, 15	fveq12d 6197	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝑧‘∪ dom 𝑧) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
24	15	fveq2d 6195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐹‘∪ dom 𝑧) = (𝐹‘∪ 𝐶))
25	24	sneqd 4189	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → {(𝐹‘∪ dom 𝑧)} = {(𝐹‘∪ 𝐶)})
26	23, 25	uneq12d 3768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) = (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
27	26	eleq1d 2686	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴 ↔ (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
28		eqidd 2623	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → ∅ = ∅)
29	27, 25, 28	ifbieq12d 4113	. . . . . 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)}, ∅))
30	23, 29	uneq12d 3768	. . . . 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)}, ∅)))
31	16, 22, 30	ifbieq12d 4113	. . . 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)}, ∅))))
32		onuni 6993	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
33	32	ad3antlr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ On)
34		sucidg 5803	. . . . . . . . 9 ⊢ (∪ 𝐶 ∈ On → ∪ 𝐶 ∈ suc ∪ 𝐶)
35	33, 34	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
36		eloni 5733	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → Ord 𝐶)
37	36	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → Ord 𝐶)
38		orduniorsuc 7030	. . . . . . . . . 10 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
40	39	orcanai 952	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
41	35, 40	eleqtrrd 2704	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
42		fvres 6207	. . . . . . 7 ⊢ (∪ 𝐶 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
43	41, 42	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
44	43	uneq1d 3766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) = ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}))
45	44	eleq1d 2686	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑧 = (𝐺 ↾ 𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((((𝐺 ↾ 𝐶)‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴 ↔ ((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴))
46	45	ifbid 4108	. . . . . 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)}, ∅))
47	43, 46	uneq12d 3768	. . . . 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)}, ∅)))
48	47	ifeq2da 4117	. . . 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)}, ∅))))
49	31, 48	eqtrd 2656	. . 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)}, ∅))))
50		fnfun 5988	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
51	7, 50	ax-mp 5	. . . 4 ⊢ Fun 𝐺
52		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
53		resfunexg 6479	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
54	51, 52, 53	sylancr 695	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
55		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
56		elex 3212	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
57	55, 56	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
58		funimaexg 5975	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
59	51, 58	mpan 706	. . . . . 6 ⊢ (𝐶 ∈ On → (𝐺 “ 𝐶) ∈ V)
60		uniexg 6955	. . . . . 6 ⊢ ((𝐺 “ 𝐶) ∈ V → ∪ (𝐺 “ 𝐶) ∈ V)
61	59, 60	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → ∪ (𝐺 “ 𝐶) ∈ V)
62		ifcl 4130	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∪ (𝐺 “ 𝐶) ∈ V) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
63	57, 61, 62	syl2an 494	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)) ∈ V)
64		fvex 6201	. . . . 5 ⊢ (𝐺‘∪ 𝐶) ∈ V
65		snex 4908	. . . . . 6 ⊢ {(𝐹‘∪ 𝐶)} ∈ V
66		0ex 4790	. . . . . 6 ⊢ ∅ ∈ V
67	65, 66	ifex 4156	. . . . 5 ⊢ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅) ∈ V
68	64, 67	unex 6956	. . . 4 ⊢ ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)) ∈ V
69		ifcl 4130	. . . 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)
70	63, 68, 69	sylancl 694	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ On) → if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅))) ∈ V)
71	4, 49, 54, 70	fvmptd 6288	. 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)}, ∅))))
72	3, 71	eqtrd 2656	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 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Ord word 5722  Oncon0 5723  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  recscrecs 7467  Fincfn 7955  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  ttukeylem4  9334  ttukeylem5  9335  ttukeylem6  9336  ttukeylem7  9337