![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3793 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 5535 | . . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘〈𝑥, 𝑦〉) = (2nd ‘〈𝐴, 𝑦〉)) |
3 | 2 | eqeq1d 2198 | . 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘〈𝑥, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝑦〉) = 𝑦)) |
4 | opeq2 3794 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
5 | 4 | fveq2d 5535 | . . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘〈𝐴, 𝑦〉) = (2nd ‘〈𝐴, 𝐵〉)) |
6 | id 19 | . . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
7 | 5, 6 | eqeq12d 2204 | . 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘〈𝐴, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝐵〉) = 𝐵)) |
8 | vex 2755 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 2755 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op2nd 6167 | . 2 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
11 | 3, 7, 10 | vtocl2g 2816 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 ‘cfv 5232 2nd c2nd 6159 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5234 df-fv 5240 df-2nd 6161 |
This theorem is referenced by: ot2ndg 6173 ot3rdgg 6174 2ndconst 6242 xpmapenlem 6872 2ndinl 7099 2ndinr 7101 mulpipq 7396 suplocexprlem2b 7738 aprcl 8628 frec2uzrdg 10435 frecuzrdgsuc 10440 eucalglt 12084 eucalg 12086 qredeu 12124 sqpweven 12202 2sqpwodd 12203 qnumdenbi 12219 upxp 14209 uptx 14211 txmetcnp 14455 |
Copyright terms: Public domain | W3C validator |