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Theorem fparlem1 7952

Description: Lemma for fpar 7956. (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 6793	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1^st ↾ (V × V))‘𝑦) = (1^st ‘𝑦))
2	1	eqeq1d 2740	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1^st ‘𝑦) = 𝑥))
3		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elsn2 4600	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ (1^st ‘𝑦) = 𝑥)
5		fvex 6787	. . . . . . 7 ⊢ (2^nd ‘𝑦) ∈ V
6	5	biantru 530	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
7	4, 6	bitr3i 276	. . . . 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 575	. . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
10		f1stres 7855	. . . 4 ⊢ (1^st ↾ (V × V)):(V × V)⟶V
11		ffn 6600	. . . 4 ⊢ ((1^st ↾ (V × V)):(V × V)⟶V → (1^st ↾ (V × V)) Fn (V × V))
12		fniniseg 6937	. . . 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 7866	. . 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 2735	1 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 {csn 4561 × cxp 5587 ^◡ccnv 5588 ↾ cres 5591 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-1st 7831 df-2nd 7832

This theorem is referenced by: fparlem3 7954

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