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Theorem fparlem1 8051

Description: Lemma for fpar 8055. (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 6850	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1^st ↾ (V × V))‘𝑦) = (1^st ‘𝑦))
2	1	eqeq1d 2735	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1^st ‘𝑦) = 𝑥))
3		vex 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elsn2 4619	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ (1^st ‘𝑦) = 𝑥)
5		fvex 6844	. . . . . . 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 7954	. . . 4 ⊢ (1^st ↾ (V × V)):(V × V)⟶V
11		ffn 6659	. . . 4 ⊢ ((1^st ↾ (V × V)):(V × V)⟶V → (1^st ↾ (V × V)) Fn (V × V))
12		fniniseg 7002	. . . 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 7965	. . 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 2730	1 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 {csn 4577 × cxp 5619 ^◡ccnv 5620 ↾ cres 5623 “ cima 5624 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 1^st c1st 7928 2^nd c2nd 7929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-1st 7930 df-2nd 7931

This theorem is referenced by: fparlem3 8053

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