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Mirrors > Home > MPE Home > Th. List > fparlem2 | Structured version Visualization version GIF version |
Description: Lemma for fpar 7674. (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 6564 | . . . . . 6 ⊢ (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd ‘𝑥)) | |
2 | 1 | eqeq1d 2799 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd ‘𝑥) = 𝑦)) |
3 | vex 3443 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elsn2 4515 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ (2nd ‘𝑥) = 𝑦) |
5 | fvex 6558 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
6 | 5 | biantrur 531 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
7 | 4, 6 | bitr3i 278 | . . . . 5 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
8 | 2, 7 | syl6bb 288 | . . . 4 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
9 | 8 | pm5.32i 575 | . . 3 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
10 | f2ndres 7577 | . . . 4 ⊢ (2nd ↾ (V × V)):(V × V)⟶V | |
11 | ffn 6389 | . . . 4 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 6702 | . . . 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 7587 | . . 3 ⊢ (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) | |
15 | 9, 13, 14 | 3bitr4i 304 | . 2 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦})) |
16 | 15 | eqriv 2794 | 1 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 {csn 4478 × cxp 5448 ◡ccnv 5449 ↾ cres 5452 “ cima 5453 Fn wfn 6227 ⟶wf 6228 ‘cfv 6232 1st c1st 7550 2nd c2nd 7551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-1st 7552 df-2nd 7553 |
This theorem is referenced by: fparlem4 7673 |
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