Theorem ptpjpre1 23456

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 6822	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → (𝑤‘𝑘) = (𝑤‘𝐼))
2		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
3	2	unieqd 4871	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
4	1, 3	eleq12d 2822	. . . . . . 7 ⊢ (𝑘 = 𝐼 → ((𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼)))
5		vex 3440	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
6	5	elixp 8831	. . . . . . . . . 10 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘)))
7	6	simprbi 496	. . . . . . . . 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 2846	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
10	9	adantl 481	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
11		simplrl 776	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → 𝐼 ∈ 𝐴)
12	4, 10, 11	rspcdva 3578	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼))
13	12	fmpttd 7049	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼))
14		ffn 6652	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) Fn 𝑋)
15		elpreima 6992	. . . . 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 6821	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤‘𝐼) = (𝑧‘𝐼))
18		eqid 2729	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))
19		fvex 6835	. . . . . . . . 9 ⊢ (𝑧‘𝐼) ∈ V
20	17, 18, 19	fvmpt 6930	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) = (𝑧‘𝐼))
21	20	eleq1d 2813	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 → (((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈 ↔ (𝑧‘𝐼) ∈ 𝑈))
22	21	pm5.32i 574	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈))
23	8	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24		vex 3440	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	24	elixp 8831	. . . . . . . . 9 ⊢ (𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
26	23, 25	bitri 275	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
27	26	anbi1i 624	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ ((𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)) ∧ (𝑧‘𝐼) ∈ 𝑈))
28		anass 468	. . . . . . 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 275	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
30	22, 29	bitri 275	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
31		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝐼) ∈ 𝑈)
32		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → (𝑧‘𝑘) = (𝑧‘𝐼))
33		iftrue 4482	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = 𝑈)
34	32, 33	eleq12d 2822	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝐼) ∈ 𝑈))
35	31, 34	syl5ibrcom 247	. . . . . . . . . . . 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 772	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))
37		iffalse 4485	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘))
38	37	eleq2d 2814	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
39	36, 38	syl5ibrcom 247	. . . . . . . . . . . 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 179	. . . . . . . . . . 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 456	. . . . . . . . . 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 3143	. . . . . . . . 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 453	. . . . . . . 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 465	. . . . . . 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 4888	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (𝐹‘𝐼) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
46	45	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
47	33, 3	sseq12d 3969	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘) ↔ 𝑈 ⊆ ∪ (𝐹‘𝐼)))
48	46, 47	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
49		ssid 3958	. . . . . . . . . . . 12 ⊢ ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘)
50	37, 49	eqsstrdi 3980	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
51	48, 50	pm2.61d1 180	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
52	51	sseld 3934	. . . . . . . . 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 3143	. . . . . . . 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 3573	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (𝑧‘𝐼) ∈ 𝑈))
55	54	ad2antrl 728	. . . . . . . 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 512	. . . . . . 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 212	. . . . . 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 630	. . . . 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 283	. . . 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 279	. . 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 8831	. . 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 289	. 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 2727	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  ifcif 4476  ∪ cuni 4858   ↦ cmpt 5173  ^◡ccnv 5618   “ cima 5622   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  Xcixp 8824  Topctop 22778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ixp 8825

This theorem is referenced by:  ptpjpre2  23465  ptbasfi  23466