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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 7864 | . 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 6784 | . . . 4 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6784 | . . . 4 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opelvv 5629 | . . 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 208 | 1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2110 Vcvv 3431 〈cop 4573 × cxp 5588 ‘cfv 6432 1st c1st 7823 2nd c2nd 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fv 6440 df-1st 7825 df-2nd 7826 |
This theorem is referenced by: wlkcpr 28006 wlkeq 28011 opfv 30991 1stpreimas 31047 ovolval2lem 44163 |
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