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 4662 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | df-pr 4564 | . . . . . 6 ⊢ {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) | |
3 | 1, 2 | eqtri 2844 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) |
4 | 3 | fveq1i 6666 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) |
5 | fvunsn 6936 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) | |
6 | 4, 5 | syl5eq 2868 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
7 | 6 | 3ad2ant3 1131 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
8 | fvsng 6937 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
9 | 8 | 3adant3 1128 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
10 | 7, 9 | eqtrd 2856 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∪ cun 3934 {csn 4561 {cpr 4563 〈cop 4567 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-res 5562 df-iota 6309 df-fun 6352 df-fv 6358 |
This theorem is referenced by: fpropnf1 7019 f1prex 7034 wrdlen2i 14298 fvpr1o 16827 linds2eq 30936 zlmodzxzscm 44398 zlmodzxzadd 44399 lincvalpr 44466 ldepspr 44521 fv2prop 44680 prelrrx2b 44694 line2ylem 44731 line2 44732 line2x 44734 line2y 44735 |
Copyright terms: Public domain | W3C validator |