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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.) (Proof shortened by BJ, 26-Sep-2024.) |
Ref | Expression |
---|---|
fvpr2g | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4735 | . . 3 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩} | |
2 | 1 | fveq1i 6891 | . 2 ⊢ ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵) |
3 | necom 2992 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvpr1g 7189 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐵 ≠ 𝐴) → ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵) = 𝐷) | |
5 | 3, 4 | syl3an3b 1403 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵) = 𝐷) |
6 | 2, 5 | eqtrid 2782 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 {cpr 4629 ⟨cop 4633 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6494 df-fun 6544 df-fv 6550 |
This theorem is referenced by: fvpr2 7194 fpropnf1 7268 f1prex 7284 wrdlen2i 14897 fvpr1o 17510 linds2eq 32771 zlmodzxzscm 47121 zlmodzxzadd 47122 lincvalpr 47186 ldepspr 47241 2arymptfv 47423 fv2prop 47473 prelrrx2b 47487 line2ylem 47524 line2 47525 line2x 47527 line2y 47528 |
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