![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fparlem1 | Structured version Visualization version GIF version |
Description: Lemma for fpar 8052. (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 6865 | . . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st ‘𝑦)) | |
2 | 1 | eqeq1d 2735 | . . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st ‘𝑦) = 𝑥)) |
3 | vex 3451 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | 3 | elsn2 4629 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ (1st ‘𝑦) = 𝑥) |
5 | fvex 6859 | . . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | biantru 531 | . . . . . 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 576 | . . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) |
10 | f1stres 7949 | . . . 4 ⊢ (1st ↾ (V × V)):(V × V)⟶V | |
11 | ffn 6672 | . . . 4 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 7014 | . . . 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 7960 | . . 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 2730 | 1 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 {csn 4590 × cxp 5635 ◡ccnv 5636 ↾ cres 5639 “ cima 5640 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 1st c1st 7923 2nd c2nd 7924 |
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 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-1st 7925 df-2nd 7926 |
This theorem is referenced by: fparlem3 8050 |
Copyright terms: Public domain | W3C validator |