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Mirrors > Home > ILE Home > Th. List > fvpr1 | GIF version |
Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Ref | Expression |
---|---|
fvpr1.1 | ⊢ 𝐴 ∈ V |
fvpr1.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fvpr1 | ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3598 | . . . 4 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | fveq1i 5511 | . . 3 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) |
3 | necom 2431 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvpr1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
5 | fvunsng 5705 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ≠ 𝐴) → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
7 | 3, 6 | sylbi 121 | . . 3 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
8 | 2, 7 | eqtrid 2222 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
9 | fvpr1.2 | . . 3 ⊢ 𝐶 ∈ V | |
10 | 4, 9 | fvsn 5706 | . 2 ⊢ ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶 |
11 | 8, 10 | eqtrdi 2226 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 Vcvv 2737 ∪ cun 3127 {csn 3591 {cpr 3592 〈cop 3594 ‘cfv 5211 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-res 4634 df-iota 5173 df-fun 5213 df-fv 5219 |
This theorem is referenced by: fvpr2 5716 |
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