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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 7730 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | eqeq1d 2825 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉)) |
3 | fvex 6685 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6685 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opth 5370 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) |
6 | 2, 5 | syl6bb 289 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: eqop2 7734 op1steq 7735 el2xptp0 7738 lsmhash 18833 txhmeo 22413 ptuncnv 22417 wlkcomp 27414 clwlkcomp 27562 f1od2 30459 esum2dlem 31353 poimirlem22 34916 rngosn3 35204 dvhb1dimN 38124 |
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