Theorem ptpjpre1 23633

Description: The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)

Hypothesis

Ref	Expression
ptpjpre1.1	⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)

Assertion

Ref	Expression
ptpjpre1	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) = X𝑘 ∈ 𝐴 if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))

Distinct variable groups:   𝑤,𝑘,𝐴   𝑘,𝐹,𝑤   𝑘,𝐼,𝑤   𝑈,𝑘,𝑤   𝑘,𝑉,𝑤   𝑤,𝑋

Allowed substitution hint:   𝑋(𝑘)

Proof of Theorem ptpjpre1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6869	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → (𝑤‘𝑘) = (𝑤‘𝐼))
2		fveq2 6869	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
3	2	unieqd 4880	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
4	1, 3	eleq12d 2858	. . . . . . 7 ⊢ (𝑘 = 𝐼 → ((𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼)))
5		vex 3460	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
6	5	elixp 8888	. . . . . . . . . 10 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘)))
7	6	simprbi 501	. . . . . . . . 9 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
8		ptpjpre1.1	. . . . . . . . 9 ⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
9	7, 8	eleq2s 2882	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
10	9	adantl 485	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
11		simplrl 786	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → 𝐼 ∈ 𝐴)
12	4, 10, 11	rspcdva 3584	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼))
13	12	fmpttd 7098	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼))
14		ffn 6693	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) Fn 𝑋)
15		elpreima 7041	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) Fn 𝑋 → (𝑧 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈)))
16	13, 14, 15	3syl 18	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑧 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈)))
17		fveq1 6868	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤‘𝐼) = (𝑧‘𝐼))
18		eqid 2764	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))
19		fvex 6882	. . . . . . . . 9 ⊢ (𝑧‘𝐼) ∈ V
20	17, 18, 19	fvmpt 6977	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) = (𝑧‘𝐼))
21	20	eleq1d 2849	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 → (((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈 ↔ (𝑧‘𝐼) ∈ 𝑈))
22	21	pm5.32i 582	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈))
23	8	eleq2i 2856	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24		vex 3460	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	24	elixp 8888	. . . . . . . . 9 ⊢ (𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
26	23, 25	bitri 277	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
27	26	anbi1i 633	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ ((𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)) ∧ (𝑧‘𝐼) ∈ 𝑈))
28		anass 472	. . . . . . 7 ⊢ (((𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)) ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
29	27, 28	bitri 277	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
30	22, 29	bitri 277	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
31		simprl 780	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝐼) ∈ 𝑈)
32		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → (𝑧‘𝑘) = (𝑧‘𝐼))
33		iftrue 4488	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = 𝑈)
34	32, 33	eleq12d 2858	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝐼) ∈ 𝑈))
35	31, 34	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑘 = 𝐼 → (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
36		simprr 782	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))
37		iffalse 4491	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘))
38	37	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
39	36, 38	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (¬ 𝑘 = 𝐼 → (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
40	35, 39	pm2.61d 180	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))
41	40	expr 460	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ (𝑧‘𝐼) ∈ 𝑈) → ((𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) → (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
42	41	ralimdv 3178	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ (𝑧‘𝐼) ∈ 𝑈) → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
43	42	expimpd 457	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (((𝑧‘𝐼) ∈ 𝑈 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)) → ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
44	43	ancomsd 469	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → ((∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈) → ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
45		elssuni 4899	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (𝐹‘𝐼) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
46	45	ad2antll 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
47	33, 3	sseq12d 3971	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘) ↔ 𝑈 ⊆ ∪ (𝐹‘𝐼)))
48	46, 47	syl5ibrcom 249	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
49		ssid 3960	. . . . . . . . . . . 12 ⊢ ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘)
50	37, 49	eqsstrdi 3982	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
51	48, 50	pm2.61d1 181	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
52	51	sseld 3937	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
53	52	ralimdv 3178	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
54	34	rspcv 3579	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (𝑧‘𝐼) ∈ 𝑈))
55	54	ad2antrl 738	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (𝑧‘𝐼) ∈ 𝑈))
56	53, 55	jcad 520	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
57	44, 56	impbid 214	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → ((∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
58	57	anbi2d 639	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → ((𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))))
59	30, 58	bitrid 285	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))))
60	16, 59	bitrd 281	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑧 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))))
61	24	elixp 8888	. . 3 ⊢ (𝑧 ∈ X𝑘 ∈ 𝐴 if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
62	60, 61	bitr4di 291	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑧 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) ↔ 𝑧 ∈ X𝑘 ∈ 𝐴 if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘))))
63	62	eqrdv 2762	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) = X𝑘 ∈ 𝐴 if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  ifcif 4482  ∪ cuni 4867   ↦ cmpt 5183  ^◡ccnv 5648   “ cima 5652   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523  Xcixp 8881  Topctop 22955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ixp 8882

This theorem is referenced by:  ptpjpre2  23642  ptbasfi  23643