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Theorem fvtp1 7215
Description: The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp1.1 𝐴 ∈ V
fvtp1.4 𝐷 ∈ V
Assertion
Ref Expression
fvtp1 ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Proof of Theorem fvtp1
StepHypRef Expression
1 df-tp 4631 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})
21fveq1i 6907 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴)
3 necom 2994 . . . 4 (𝐴𝐶𝐶𝐴)
4 fvunsn 7199 . . . 4 (𝐶𝐴 → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴))
53, 4sylbi 217 . . 3 (𝐴𝐶 → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴))
6 fvtp1.1 . . . 4 𝐴 ∈ V
7 fvtp1.4 . . . 4 𝐷 ∈ V
86, 7fvpr1 7212 . . 3 (𝐴𝐵 → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴) = 𝐷)
95, 8sylan9eqr 2799 . 2 ((𝐴𝐵𝐴𝐶) → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = 𝐷)
102, 9eqtrid 2789 1 ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cun 3949  {csn 4626  {cpr 4628  {ctp 4630  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-tp 4631  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:  fvtp2  7216  fntpb  7229  rabren3dioph  42826  nnsum4primesodd  47783  nnsum4primesoddALTV  47784
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