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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 3540 | . . . . 5 ⊢ (𝑎 = A → 〈𝑎, 𝑏〉 = 〈A, 𝑏〉) | |
2 | 1 | sneqd 3380 | . . . 4 ⊢ (𝑎 = A → {〈𝑎, 𝑏〉} = {〈A, 𝑏〉}) |
3 | id 19 | . . . 4 ⊢ (𝑎 = A → 𝑎 = A) | |
4 | 2, 3 | fveq12d 5127 | . . 3 ⊢ (𝑎 = A → ({〈𝑎, 𝑏〉}‘𝑎) = ({〈A, 𝑏〉}‘A)) |
5 | 4 | eqeq1d 2045 | . 2 ⊢ (𝑎 = A → (({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 ↔ ({〈A, 𝑏〉}‘A) = 𝑏)) |
6 | opeq2 3541 | . . . . 5 ⊢ (𝑏 = B → 〈A, 𝑏〉 = 〈A, B〉) | |
7 | 6 | sneqd 3380 | . . . 4 ⊢ (𝑏 = B → {〈A, 𝑏〉} = {〈A, B〉}) |
8 | 7 | fveq1d 5123 | . . 3 ⊢ (𝑏 = B → ({〈A, 𝑏〉}‘A) = ({〈A, B〉}‘A)) |
9 | id 19 | . . 3 ⊢ (𝑏 = B → 𝑏 = B) | |
10 | 8, 9 | eqeq12d 2051 | . 2 ⊢ (𝑏 = B → (({〈A, 𝑏〉}‘A) = 𝑏 ↔ ({〈A, B〉}‘A) = B)) |
11 | vex 2554 | . . 3 ⊢ 𝑎 ∈ V | |
12 | vex 2554 | . . 3 ⊢ 𝑏 ∈ V | |
13 | 11, 12 | fvsn 5301 | . 2 ⊢ ({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 |
14 | 5, 10, 13 | vtocl2g 2611 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ({〈A, B〉}‘A) = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 {csn 3367 〈cop 3370 ‘cfv 4845 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: fsnunfv 5306 fvpr1g 5310 fvpr2g 5311 tfr0 5878 fseq1p1m1 8726 1fv 8766 |
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