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Theorem fvtp3 6970
Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp3.1 𝐶 ∈ V
fvtp3.4 𝐹 ∈ V
Assertion
Ref Expression
fvtp3 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Proof of Theorem fvtp3
StepHypRef Expression
1 tprot 4641 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}
21fveq1i 6676 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶)
3 necom 2987 . . . 4 (𝐴𝐶𝐶𝐴)
4 fvtp3.1 . . . . 5 𝐶 ∈ V
5 fvtp3.4 . . . . 5 𝐹 ∈ V
64, 5fvtp2 6969 . . . 4 ((𝐵𝐶𝐶𝐴) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
73, 6sylan2b 597 . . 3 ((𝐵𝐶𝐴𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
87ancoms 462 . 2 ((𝐴𝐶𝐵𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
92, 8syl5eq 2785 1 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  wne 2934  Vcvv 3398  {ctp 4521  cop 4523  cfv 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-res 5538  df-iota 6298  df-fun 6342  df-fv 6348
This theorem is referenced by:  fntpb  6983  rabren3dioph  40201  nnsum4primesodd  44774  nnsum4primesoddALTV  44775
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