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Theorem fparlem1 8136

Description: Lemma for fpar 8140. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
fparlem1	⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Proof of Theorem fparlem1 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 6926	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1^st ↾ (V × V))‘𝑦) = (1^st ‘𝑦))
2	1	eqeq1d 2737	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1^st ‘𝑦) = 𝑥))
3		vex 3482	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elsn2 4670	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ (1^st ‘𝑦) = 𝑥)
5		fvex 6920	. . . . . . 7 ⊢ (2^nd ‘𝑦) ∈ V
6	5	biantru 529	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
7	4, 6	bitr3i 277	. . . . 5 ⊢ ((1^st ‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
8	2, 7	bitrdi 287	. . . 4 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
9	8	pm5.32i 574	. . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
10		f1stres 8037	. . . 4 ⊢ (1^st ↾ (V × V)):(V × V)⟶V
11		ffn 6737	. . . 4 ⊢ ((1^st ↾ (V × V)):(V × V)⟶V → (1^st ↾ (V × V)) Fn (V × V))
12		fniniseg 7080	. . . 4 ⊢ ((1^st ↾ (V × V)) Fn (V × V) → (𝑦 ∈ (^◡(1^st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥)))
13	10, 11, 12	mp2b 10	. . 3 ⊢ (𝑦 ∈ (^◡(1^st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥))
14		elxp7 8048	. . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
15	9, 13, 14	3bitr4i 303	. 2 ⊢ (𝑦 ∈ (^◡(1^st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V))
16	15	eqriv 2732	1 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 × cxp 5687 ^◡ccnv 5688 ↾ cres 5691 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 1^st c1st 8011 2^nd c2nd 8012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1st 8013 df-2nd 8014

This theorem is referenced by: fparlem3 8138

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