![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fparlem2 | Structured version Visualization version GIF version |
Description: Lemma for fpar 8097. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fparlem2 | ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6907 | . . . . . 6 ⊢ (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd ‘𝑥)) | |
2 | 1 | eqeq1d 2735 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd ‘𝑥) = 𝑦)) |
3 | vex 3479 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elsn2 4666 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ (2nd ‘𝑥) = 𝑦) |
5 | fvex 6901 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
6 | 5 | biantrur 532 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
7 | 4, 6 | bitr3i 277 | . . . . 5 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
8 | 2, 7 | bitrdi 287 | . . . 4 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
9 | 8 | pm5.32i 576 | . . 3 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
10 | f2ndres 7995 | . . . 4 ⊢ (2nd ↾ (V × V)):(V × V)⟶V | |
11 | ffn 6714 | . . . 4 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 7057 | . . . 4 ⊢ ((2nd ↾ (V × V)) Fn (V × V) → (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦))) | |
13 | 10, 11, 12 | mp2b 10 | . . 3 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦)) |
14 | elxp7 8005 | . . 3 ⊢ (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) | |
15 | 9, 13, 14 | 3bitr4i 303 | . 2 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦})) |
16 | 15 | eqriv 2730 | 1 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4627 × cxp 5673 ◡ccnv 5674 ↾ cres 5677 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 1st c1st 7968 2nd c2nd 7969 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-1st 7970 df-2nd 7971 |
This theorem is referenced by: fparlem4 8096 |
Copyright terms: Public domain | W3C validator |