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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 3635 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | df-pr 3567 | . . . . . 6 ⊢ {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) | |
3 | 1, 2 | eqtri 2178 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) |
4 | 3 | fveq1i 5469 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) |
5 | fvunsng 5661 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) | |
6 | 4, 5 | syl5eq 2202 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
7 | 6 | 3adant2 1001 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
8 | fvsng 5663 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
9 | 8 | 3adant3 1002 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
10 | 7, 9 | eqtrd 2190 | 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 963 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 ∪ cun 3100 {csn 3560 {cpr 3561 〈cop 3563 ‘cfv 5170 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-res 4598 df-iota 5135 df-fun 5172 df-fv 5178 |
This theorem is referenced by: (None) |
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