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Theorem fntp 6441
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 6437 . 2 ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
84, 5, 6dmtpop 6081 . 2 dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}
9 df-fn 6383 . 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 593 1 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  wne 2940  Vcvv 3408  {ctp 4545  cop 4547  dom cdm 5551  Fun wfun 6374   Fn wfn 6375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-fun 6382  df-fn 6383
This theorem is referenced by:  fntpb  7025  rabren3dioph  40340
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