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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 7096 | . 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 6113 | . . . 4 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6113 | . . . 4 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opelvv 5088 | . . 3 ⊢ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (V × V) |
6 | 2, 5 | syl6eqel 2696 | . 2 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → 𝐴 ∈ (V × V)) |
7 | 1, 6 | impbii 198 | 1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: 2wlkeq 26235 opfv 28828 1stpreimas 28866 ovolval2lem 39533 1wlkcpr 40833 1wlkeq 40838 |
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