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Theorem fparlem1 7880

Description: Lemma for fpar 7884. (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 6736	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1^st ↾ (V × V))‘𝑦) = (1^st ‘𝑦))
2	1	eqeq1d 2739	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1^st ‘𝑦) = 𝑥))
3		vex 3412	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elsn2 4580	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ (1^st ‘𝑦) = 𝑥)
5		fvex 6730	. . . . . . 7 ⊢ (2^nd ‘𝑦) ∈ V
6	5	biantru 533	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
7	4, 6	bitr3i 280	. . . . 5 ⊢ ((1^st ‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
8	2, 7	bitrdi 290	. . . 4 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
9	8	pm5.32i 578	. . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
10		f1stres 7785	. . . 4 ⊢ (1^st ↾ (V × V)):(V × V)⟶V
11		ffn 6545	. . . 4 ⊢ ((1^st ↾ (V × V)):(V × V)⟶V → (1^st ↾ (V × V)) Fn (V × V))
12		fniniseg 6880	. . . 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 7796	. . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
15	9, 13, 14	3bitr4i 306	. 2 ⊢ (𝑦 ∈ (^◡(1^st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V))
16	15	eqriv 2734	1 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 {csn 4541 × cxp 5549 ^◡ccnv 5550 ↾ cres 5553 “ cima 5554 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 1^st c1st 7759 2^nd c2nd 7760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-1st 7761 df-2nd 7762

This theorem is referenced by: fparlem3 7882

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