![]() |
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 3790 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
2 | 1 | sneqd 3617 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
3 | id 19 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | 2, 3 | fveq12d 5534 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}‘𝑎) = ({〈𝐴, 𝑏〉}‘𝐴)) |
5 | 4 | eqeq1d 2196 | . 2 ⊢ (𝑎 = 𝐴 → (({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 ↔ ({〈𝐴, 𝑏〉}‘𝐴) = 𝑏)) |
6 | opeq2 3791 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | sneqd 3617 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
8 | 7 | fveq1d 5529 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
9 | id 19 | . . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
10 | 8, 9 | eqeq12d 2202 | . 2 ⊢ (𝑏 = 𝐵 → (({〈𝐴, 𝑏〉}‘𝐴) = 𝑏 ↔ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) |
11 | vex 2752 | . . 3 ⊢ 𝑎 ∈ V | |
12 | vex 2752 | . . 3 ⊢ 𝑏 ∈ V | |
13 | 11, 12 | fvsn 5724 | . 2 ⊢ ({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 |
14 | 5, 10, 13 | vtocl2g 2813 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 {csn 3604 〈cop 3607 ‘cfv 5228 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 |
This theorem is referenced by: fsnunfv 5730 fvpr1g 5735 fvpr2g 5736 tfr0dm 6337 fseq1p1m1 10108 1fv 10153 sumsnf 11431 prodsnf 11614 setsslid 12527 mgm1 12808 sgrp1 12836 mnd1 12869 mnd1id 12870 grp1 13003 ring1 13309 ixpsnbasval 13655 |
Copyright terms: Public domain | W3C validator |