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Mirrors > Home > MPE Home > Th. List > eqop | Structured version Visualization version GIF version |
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
eqop | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 7440 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | eqeq1d 2801 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉)) |
3 | fvex 6424 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6424 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opth 5135 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) |
6 | 2, 5 | syl6bb 279 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 〈cop 4374 × cxp 5310 ‘cfv 6101 1st c1st 7399 2nd c2nd 7400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fv 6109 df-1st 7401 df-2nd 7402 |
This theorem is referenced by: eqop2 7444 op1steq 7445 el2xptp0 7447 lsmhash 18431 txhmeo 21935 ptuncnv 21939 wlkcomp 26880 clwlkcomp 27033 f1od2 30017 esum2dlem 30670 poimirlem22 33920 rngosn3 34210 dvhb1dimN 37007 |
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