Theorem ptpjpre1 23699

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 6884	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → (𝑤‘𝑘) = (𝑤‘𝐼))
2		fveq2 6884	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
3	2	unieqd 4889	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
4	1, 3	eleq12d 2863	. . . . . . 7 ⊢ (𝑘 = 𝐼 → ((𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼)))
5		vex 3467	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
6	5	elixp 8904	. . . . . . . . . 10 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘)))
7	6	simprbi 502	. . . . . . . . 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 2887	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
10	9	adantl 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 (𝑤‘𝑘) ∈ ∪ (𝐹‘𝑘))
11		simplrl 788	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → 𝐼 ∈ 𝐴)
12	4, 10, 11	rspcdva 3591	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼))
13	12	fmpttd 7113	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼))
14		ffn 6708	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) Fn 𝑋)
15		elpreima 7056	. . . . 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 19	. . . 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 6883	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤‘𝐼) = (𝑧‘𝐼))
18		eqid 2769	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))
19		fvex 6897	. . . . . . . . 9 ⊢ (𝑧‘𝐼) ∈ V
20	17, 18, 19	fvmpt 6992	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) = (𝑧‘𝐼))
21	20	eleq1d 2854	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 → (((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈 ↔ (𝑧‘𝐼) ∈ 𝑈))
22	21	pm5.32i 584	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈))
23	8	eleq2i 2861	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24		vex 3467	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	24	elixp 8904	. . . . . . . . 9 ⊢ (𝑧 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
26	23, 25	bitri 278	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 ↔ (𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
27	26	anbi1i 635	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ ((𝑧 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)) ∧ (𝑧‘𝐼) ∈ 𝑈))
28		anass 473	. . . . . . 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 278	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑧‘𝐼) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
30	22, 29	bitri 278	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))‘𝑧) ∈ 𝑈) ↔ (𝑧 Fn 𝐴 ∧ (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘) ∧ (𝑧‘𝐼) ∈ 𝑈)))
31		simprl 782	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝐼) ∈ 𝑈)
32		fveq2 6884	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → (𝑧‘𝑘) = (𝑧‘𝐼))
33		iftrue 4498	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = 𝑈)
34	32, 33	eleq12d 2863	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝐼) ∈ 𝑈))
35	31, 34	syl5ibrcom 250	. . . . . . . . . . . 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 784	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ ((𝑧‘𝐼) ∈ 𝑈 ∧ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))) → (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘))
37		iffalse 4501	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘))
38	37	eleq2d 2855	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐼 → ((𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ↔ (𝑧‘𝑘) ∈ ∪ (𝐹‘𝑘)))
39	36, 38	syl5ibrcom 250	. . . . . . . . . . . 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 181	. . . . . . . . . . 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 461	. . . . . . . . . 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 3185	. . . . . . . . 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 458	. . . . . . . 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 470	. . . . . . 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 4908	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (𝐹‘𝐼) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
46	45	ad2antll 741	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → 𝑈 ⊆ ∪ (𝐹‘𝐼))
47	33, 3	sseq12d 3978	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘) ↔ 𝑈 ⊆ ∪ (𝐹‘𝐼)))
48	46, 47	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
49		ssid 3967	. . . . . . . . . . . 12 ⊢ ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘)
50	37, 49	eqsstrdi 3989	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐼 → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
51	48, 50	pm2.61d1 182	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
52	51	sseld 3944	. . . . . . . . 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 3185	. . . . . . . 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 3586	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑧‘𝑘) ∈ if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)) → (𝑧‘𝐼) ∈ 𝑈))
55	54	ad2antrl 740	. . . . . . . 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 521	. . . . . . 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 215	. . . . . 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 641	. . . . 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 286	. . . 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 282	. . 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 8904	. . 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 292	. 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 2767	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ifcif 4492  ∪ cuni 4876   ↦ cmpt 5196  ^◡ccnv 5663   “ cima 5667   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  Xcixp 8897  Topctop 23021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ixp 8898

This theorem is referenced by:  ptpjpre2  23708  ptbasfi  23709