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Theorem fvpr1g 6499
Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
Assertion
Ref Expression
fvpr1g ((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Proof of Theorem fvpr1g
StepHypRef Expression
1 df-pr 4213 . . . . 5 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21fveq1i 6230 . . . 4 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴)
3 necom 2876 . . . . 5 (𝐴𝐵𝐵𝐴)
4 fvunsn 6486 . . . . 5 (𝐵𝐴 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
53, 4sylbi 207 . . . 4 (𝐴𝐵 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
62, 5syl5eq 2697 . . 3 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
763ad2ant3 1104 . 2 ((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
8 fvsng 6488 . . 3 ((𝐴𝑉𝐶𝑊) → ({⟨𝐴, 𝐶⟩}‘𝐴) = 𝐶)
983adant3 1101 . 2 ((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩}‘𝐴) = 𝐶)
107, 9eqtrd 2685 1 ((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cun 3605  {csn 4210  {cpr 4212  cop 4216  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  fvtp1g  6504  fpropnf1  6564  f1prex  6579  wrdlen2i  13732  zlmodzxzscm  42460  zlmodzxzadd  42461  lincvalpr  42532  ldepspr  42587
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