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 3700 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
2 | 1 | sneqd 3535 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
3 | id 19 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | 2, 3 | fveq12d 5421 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}‘𝑎) = ({〈𝐴, 𝑏〉}‘𝐴)) |
5 | 4 | eqeq1d 2146 | . 2 ⊢ (𝑎 = 𝐴 → (({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 ↔ ({〈𝐴, 𝑏〉}‘𝐴) = 𝑏)) |
6 | opeq2 3701 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | sneqd 3535 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
8 | 7 | fveq1d 5416 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
9 | id 19 | . . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
10 | 8, 9 | eqeq12d 2152 | . 2 ⊢ (𝑏 = 𝐵 → (({〈𝐴, 𝑏〉}‘𝐴) = 𝑏 ↔ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) |
11 | vex 2684 | . . 3 ⊢ 𝑎 ∈ V | |
12 | vex 2684 | . . 3 ⊢ 𝑏 ∈ V | |
13 | 11, 12 | fvsn 5608 | . 2 ⊢ ({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 |
14 | 5, 10, 13 | vtocl2g 2745 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3522 〈cop 3525 ‘cfv 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 |
This theorem is referenced by: fsnunfv 5614 fvpr1g 5619 fvpr2g 5620 tfr0dm 6212 fseq1p1m1 9867 1fv 9909 sumsnf 11171 setsslid 11998 |
Copyright terms: Public domain | W3C validator |