Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fvpr2g | Structured version Visualization version 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 4211 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | df-pr 4128 | . . . . . 6 ⊢ {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) | |
3 | 1, 2 | eqtri 2632 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) |
4 | 3 | fveq1i 6104 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) |
5 | fvunsn 6350 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) | |
6 | 4, 5 | syl5eq 2656 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
7 | 6 | 3ad2ant3 1077 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
8 | fvsng 6352 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
9 | 8 | 3adant3 1074 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
10 | 7, 9 | eqtrd 2644 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∪ cun 3538 {csn 4125 {cpr 4127 〈cop 4131 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: f1prex 6439 wrdlen2i 13534 constr1trl 26118 1pthon 26121 constr3lem4 26175 fpropnf1 40337 zlmodzxzscm 41928 zlmodzxzadd 41929 lincvalpr 42001 ldepspr 42056 |
Copyright terms: Public domain | W3C validator |