fparlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fparlem1		Structured version Visualization version GIF version

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 6846	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1^st ↾ (V × V))‘𝑦) = (1^st ‘𝑦))
2	1	eqeq1d 2741	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1^st ‘𝑦) = 𝑥))
3		vex 3435	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elsn2 4597	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ (1^st ‘𝑦) = 𝑥)
5		fvex 6840	. . . . . . 7 ⊢ (2^nd ‘𝑦) ∈ V
6	5	biantru 534	. . . . . 6 ⊢ ((1^st ‘𝑦) ∈ {𝑥} ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
7	4, 6	bitr3i 278	. . . . 5 ⊢ ((1^st ‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V))
8	2, 7	bitrdi 288	. . . 4 ⊢ (𝑦 ∈ (V × V) → (((1^st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
9	8	pm5.32i 579	. . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1^st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
10		f1stres 7955	. . . 4 ⊢ (1^st ↾ (V × V)):(V × V)⟶V
11		ffn 6655	. . . 4 ⊢ ((1^st ↾ (V × V)):(V × V)⟶V → (1^st ↾ (V × V)) Fn (V × V))
12		fniniseg 7001	. . . 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 7966	. . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1^st ‘𝑦) ∈ {𝑥} ∧ (2^nd ‘𝑦) ∈ V)))
15	9, 13, 14	3bitr4i 304	. 2 ⊢ (𝑦 ∈ (^◡(1^st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V))
16	15	eqriv 2736	1 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 {csn 4555 × cxp 5616 ^◡ccnv 5617 ↾ cres 5620 “ cima 5621 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 1^st c1st 7929 2^nd c2nd 7930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fv 6493 df-1st 7931 df-2nd 7932

This theorem is referenced by: fparlem3 8053

W3C validator