![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1st2ndb | Structured version Visualization version GIF version |
Description: Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.) |
Ref | Expression |
---|---|
1st2ndb | ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 7710 | . 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | id 22 | . . 3 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
3 | fvex 6658 | . . . 4 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6658 | . . . 4 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opelvv 5558 | . . 3 ⊢ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (V × V) |
6 | 2, 5 | eqeltrdi 2898 | . 2 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → 𝐴 ∈ (V × V)) |
7 | 1, 6 | impbii 212 | 1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: wlkcpr 27418 wlkeq 27423 opfv 30407 1stpreimas 30465 ovolval2lem 43282 |
Copyright terms: Public domain | W3C validator |