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Mirrors > Home > ILE Home > Th. List > op2ndg | GIF version |
Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Ref | Expression |
---|---|
op2ndg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3549 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 5182 | . . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘〈𝑥, 𝑦〉) = (2nd ‘〈𝐴, 𝑦〉)) |
3 | 2 | eqeq1d 2048 | . 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘〈𝑥, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝑦〉) = 𝑦)) |
4 | opeq2 3550 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
5 | 4 | fveq2d 5182 | . . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘〈𝐴, 𝑦〉) = (2nd ‘〈𝐴, 𝐵〉)) |
6 | id 19 | . . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
7 | 5, 6 | eqeq12d 2054 | . 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘〈𝐴, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝐵〉) = 𝐵)) |
8 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 2560 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op2nd 5774 | . 2 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
11 | 3, 7, 10 | vtocl2g 2617 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 〈cop 3378 ‘cfv 4902 2nd c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-2nd 5768 |
This theorem is referenced by: ot2ndg 5780 ot3rdgg 5781 2ndconst 5843 mulpipq 6470 frec2uzrdg 9195 frecuzrdgsuc 9201 |
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