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Mirrors > Home > ILE Home > Th. List > eqop | 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 6081 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | eqeq1d 2149 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉)) |
3 | 1stexg 6073 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (1st ‘𝐴) ∈ V) | |
4 | 2ndexg 6074 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (2nd ‘𝐴) ∈ V) | |
5 | opthg 4168 | . . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
6 | 3, 4, 5 | syl2anc 409 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
7 | 2, 6 | bitrd 187 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 × cxp 4545 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: eqop2 6084 op1steq 6085 f1od2 6140 txhmeo 12527 |
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