![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fparlem1 | Structured version Visualization version GIF version |
Description: Lemma for fpar 8099. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fparlem1 | ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6903 | . . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st ‘𝑦)) | |
2 | 1 | eqeq1d 2728 | . . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st ‘𝑦) = 𝑥)) |
3 | vex 3472 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | 3 | elsn2 4662 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ (1st ‘𝑦) = 𝑥) |
5 | fvex 6897 | . . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | biantru 529 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) |
7 | 4, 6 | bitr3i 277 | . . . . 5 ⊢ ((1st ‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) |
8 | 2, 7 | bitrdi 287 | . . . 4 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) |
9 | 8 | pm5.32i 574 | . . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) |
10 | f1stres 7995 | . . . 4 ⊢ (1st ↾ (V × V)):(V × V)⟶V | |
11 | ffn 6710 | . . . 4 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 7054 | . . . 4 ⊢ ((1st ↾ (V × V)) Fn (V × V) → (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥))) | |
13 | 10, 11, 12 | mp2b 10 | . . 3 ⊢ (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥)) |
14 | elxp7 8006 | . . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) | |
15 | 9, 13, 14 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V)) |
16 | 15 | eqriv 2723 | 1 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 × cxp 5667 ◡ccnv 5668 ↾ cres 5671 “ cima 5672 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 1st c1st 7969 2nd c2nd 7970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: fparlem3 8097 |
Copyright terms: Public domain | W3C validator |