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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 6170 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | eqeq1d 2186 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉)) |
3 | 1stexg 6162 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (1st ‘𝐴) ∈ V) | |
4 | 2ndexg 6163 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (2nd ‘𝐴) ∈ V) | |
5 | opthg 4235 | . . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
6 | 3, 4, 5 | syl2anc 411 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
7 | 2, 6 | bitrd 188 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3594 × cxp 4621 ‘cfv 5212 1st c1st 6133 2nd c2nd 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fo 5218 df-fv 5220 df-1st 6135 df-2nd 6136 |
This theorem is referenced by: eqop2 6173 op1steq 6174 f1od2 6230 txhmeo 13486 |
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