Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fvsng | GIF version |
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
Ref | Expression |
---|---|
fvsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3549 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
2 | 1 | sneqd 3388 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
3 | id 19 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | 2, 3 | fveq12d 5184 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}‘𝑎) = ({〈𝐴, 𝑏〉}‘𝐴)) |
5 | 4 | eqeq1d 2048 | . 2 ⊢ (𝑎 = 𝐴 → (({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 ↔ ({〈𝐴, 𝑏〉}‘𝐴) = 𝑏)) |
6 | opeq2 3550 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | sneqd 3388 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
8 | 7 | fveq1d 5180 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
9 | id 19 | . . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
10 | 8, 9 | eqeq12d 2054 | . 2 ⊢ (𝑏 = 𝐵 → (({〈𝐴, 𝑏〉}‘𝐴) = 𝑏 ↔ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) |
11 | vex 2560 | . . 3 ⊢ 𝑎 ∈ V | |
12 | vex 2560 | . . 3 ⊢ 𝑏 ∈ V | |
13 | 11, 12 | fvsn 5358 | . 2 ⊢ ({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 |
14 | 5, 10, 13 | vtocl2g 2617 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 〈cop 3378 ‘cfv 4902 |
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-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 |
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-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 |
This theorem is referenced by: fsnunfv 5363 fvpr1g 5367 fvpr2g 5368 tfr0 5937 fseq1p1m1 8956 1fv 8996 |
Copyright terms: Public domain | W3C validator |