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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 4732 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) |
| 3 | necom 2994 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
| 4 | fvpr1g 7210 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐵 ≠ 𝐴) → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) | |
| 5 | 3, 4 | syl3an3b 1407 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) |
| 6 | 2, 5 | eqtrid 2789 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {cpr 4628 〈cop 4632 ‘cfv 6561 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fvpr2 7213 fpropnf1 7287 f1prex 7304 wrdlen2i 14981 fvpr1o 17605 linds2eq 33409 zlmodzxzscm 48273 zlmodzxzadd 48274 lincvalpr 48335 ldepspr 48390 2arymptfv 48571 fv2prop 48621 prelrrx2b 48635 line2ylem 48672 line2 48673 line2x 48675 line2y 48676 |
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