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| 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 8053 | . 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 6919 | . . . 4 ⊢ (1st ‘𝐴) ∈ V | |
| 4 | fvex 6919 | . . . 4 ⊢ (2nd ‘𝐴) ∈ V | |
| 5 | 3, 4 | opelvv 5725 | . . 3 ⊢ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (V × V) |
| 6 | 2, 5 | eqeltrdi 2849 | . 2 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → 𝐴 ∈ (V × V)) |
| 7 | 1, 6 | impbii 209 | 1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: wlkcpr 29647 wlkeq 29652 opfv 32654 1stpreimas 32715 ovolval2lem 46658 tposideq 48788 fuco22a 49045 |
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