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| Mirrors > Home > MPE Home > Th. List > op1steq | Structured version Visualization version GIF version | ||
| Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) | 
| Ref | Expression | 
|---|---|
| op1steq | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpss 5701 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
| 2 | 1 | sseli 3979 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) | 
| 3 | eqid 2737 | . . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴) | |
| 4 | eqopi 8050 | . . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) | |
| 5 | 3, 4 | mpanr2 704 | . . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) | 
| 6 | fvex 6919 | . . . . . 6 ⊢ (2nd ‘𝐴) ∈ V | |
| 7 | opeq2 4874 | . . . . . . 7 ⊢ (𝑥 = (2nd ‘𝐴) → 〈𝐵, 𝑥〉 = 〈𝐵, (2nd ‘𝐴)〉) | |
| 8 | 7 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈𝐵, (2nd ‘𝐴)〉)) | 
| 9 | 6, 8 | spcev 3606 | . . . . 5 ⊢ (𝐴 = 〈𝐵, (2nd ‘𝐴)〉 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉) | 
| 10 | 5, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉) | 
| 11 | 10 | ex 412 | . . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | 
| 12 | eqop 8056 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥))) | |
| 13 | simpl 482 | . . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵) | |
| 14 | 12, 13 | biimtrdi 253 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) | 
| 15 | 14 | exlimdv 1933 | . . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) | 
| 16 | 11, 15 | impbid 212 | . 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | 
| 17 | 2, 16 | syl 17 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: releldm2 8068 | 
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