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Theorem fvtp2g 6963
Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
Assertion
Ref Expression
fvtp2g (((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)

Proof of Theorem fvtp2g
StepHypRef Expression
1 tprot 4687 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}
21fveq1i 6673 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵)
3 necom 3071 . . . 4 (𝐴𝐵𝐵𝐴)
4 fvtp1g 6962 . . . . . 6 (((𝐵𝑉𝐸𝑊) ∧ (𝐵𝐶𝐵𝐴)) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵) = 𝐸)
54expcom 416 . . . . 5 ((𝐵𝐶𝐵𝐴) → ((𝐵𝑉𝐸𝑊) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵) = 𝐸))
65ancoms 461 . . . 4 ((𝐵𝐴𝐵𝐶) → ((𝐵𝑉𝐸𝑊) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵) = 𝐸))
73, 6sylanb 583 . . 3 ((𝐴𝐵𝐵𝐶) → ((𝐵𝑉𝐸𝑊) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵) = 𝐸))
87impcom 410 . 2 (((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐵) = 𝐸)
92, 8syl5eq 2870 1 (((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wne 3018  {ctp 4573  cop 4575  cfv 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365
This theorem is referenced by:  fvtp3g  6964
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