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Mirrors > Home > ILE Home > Th. List > fvpr2g | GIF version |
Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Ref | Expression |
---|---|
fvpr2g | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3659 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | df-pr 3590 | . . . . . 6 ⊢ {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) | |
3 | 1, 2 | eqtri 2191 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) |
4 | 3 | fveq1i 5497 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) |
5 | fvunsng 5690 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) | |
6 | 4, 5 | eqtrid 2215 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
7 | 6 | 3adant2 1011 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
8 | fvsng 5692 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
9 | 8 | 3adant3 1012 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
10 | 7, 9 | eqtrd 2203 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∪ cun 3119 {csn 3583 {cpr 3584 〈cop 3586 ‘cfv 5198 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-iota 5160 df-fun 5200 df-fv 5206 |
This theorem is referenced by: (None) |
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