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Theorem ptpjpre1 23491

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 6840	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → (𝑤‘𝑘) = (𝑤‘𝐼))
2		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
3	2	unieqd 4880	. . . . . . . 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 3448	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
6	5	elixp 8854	. . . . . . . . . 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 3586	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) ∧ 𝑤 ∈ 𝑋) → (𝑤‘𝐼) ∈ ∪ (𝐹‘𝐼))
13	12	fmpttd 7069	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼))
14		ffn 6670	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)):𝑋⟶∪ (𝐹‘𝐼) → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) Fn 𝑋)
15		elpreima 7012	. . . . 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 6839	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤‘𝐼) = (𝑧‘𝐼))
18		eqid 2729	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼))
19		fvex 6853	. . . . . . . . 9 ⊢ (𝑧‘𝐼) ∈ V
20	17, 18, 19	fvmpt 6950	. . . . . . . 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 3448	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	24	elixp 8854	. . . . . . . . 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 6840	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐼 → (𝑧‘𝑘) = (𝑧‘𝐼))
33		iftrue 4490	. . . . . . . . . . . . . 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 4493	. . . . . . . . . . . . . 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 3147	. . . . . . . . 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 4897	. . . . . . . . . . . . 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 3977	. . . . . . . . . . . 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 3966	. . . . . . . . . . . 12 ⊢ ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘)
50	37, 49	eqsstrdi 3988	. . . . . . . . . . 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 3942	. . . . . . . . 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 3147	. . . . . . . 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 3581	. . . . . . . . 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 8854	. . 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)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3911 ifcif 4484 ∪ cuni 4867 ↦ cmpt 5183 ^◡ccnv 5630 “ cima 5634 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 Xcixp 8847 Topctop 22813

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 5246 ax-nul 5256 ax-pr 5382

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 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-fv 6507 df-ixp 8848

This theorem is referenced by: ptpjpre2 23500 ptbasfi 23501

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