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Theorem fvpr2g 7138
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.)
Assertion
Ref Expression
fvpr2g ((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Proof of Theorem fvpr2g
StepHypRef Expression
1 prcom 4694 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}
21fveq1i 6844 . 2 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵)
3 necom 2994 . . 3 (𝐴𝐵𝐵𝐴)
4 fvpr1g 7137 . . 3 ((𝐵𝑉𝐷𝑊𝐵𝐴) → ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵) = 𝐷)
53, 4syl3an3b 1406 . 2 ((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐵, 𝐷⟩, ⟨𝐴, 𝐶⟩}‘𝐵) = 𝐷)
62, 5eqtrid 2785 1 ((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2940  {cpr 4589  cop 4593  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505
This theorem is referenced by:  fvpr2  7142  fpropnf1  7215  f1prex  7231  wrdlen2i  14837  fvpr1o  17447  linds2eq  32216  zlmodzxzscm  46519  zlmodzxzadd  46520  lincvalpr  46585  ldepspr  46640  2arymptfv  46822  fv2prop  46872  prelrrx2b  46886  line2ylem  46923  line2  46924  line2x  46926  line2y  46927
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