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Theorem fntp 6562
Description: A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
fntp.1 𝐴 ∈ V
fntp.2 𝐵 ∈ V
fntp.3 𝐶 ∈ V
fntp.4 𝐷 ∈ V
fntp.5 𝐸 ∈ V
fntp.6 𝐹 ∈ V
Assertion
Ref Expression
fntp ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})

Proof of Theorem fntp
StepHypRef Expression
1 fntp.1 . . 3 𝐴 ∈ V
2 fntp.2 . . 3 𝐵 ∈ V
3 fntp.3 . . 3 𝐶 ∈ V
4 fntp.4 . . 3 𝐷 ∈ V
5 fntp.5 . . 3 𝐸 ∈ V
6 fntp.6 . . 3 𝐹 ∈ V
71, 2, 3, 4, 5, 6funtp 6558 . 2 ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
84, 5, 6dmtpop 6170 . 2 dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}
9 df-fn 6499 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶} ↔ (Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ∧ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}))
107, 8, 9sylanblrc 590 1 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  {ctp 4590  cop 4592  dom cdm 5633  Fun wfun 6490   Fn wfn 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-fun 6498  df-fn 6499
This theorem is referenced by:  fntpb  7158  rabren3dioph  41115
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