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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 7205 | . 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 6201 | . . . 4 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6201 | . . . 4 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opelvv 5166 | . . 3 ⊢ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (V × V) |
6 | 2, 5 | syl6eqel 2709 | . 2 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → 𝐴 ∈ (V × V)) |
7 | 1, 6 | impbii 199 | 1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 × cxp 5112 ‘cfv 5888 1st c1st 7166 2nd c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: wlkcpr 26524 wlkeq 26529 opfv 29448 1stpreimas 29483 ovolval2lem 40857 |
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