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Theorem fpm 7776
 Description: A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
elmap.1 𝐴 ∈ V
elmap.2 𝐵 ∈ V
Assertion
Ref Expression
fpm (𝐹:𝐴𝐵𝐹 ∈ (𝐵pm 𝐴))

Proof of Theorem fpm
StepHypRef Expression
1 elmap.1 . 2 𝐴 ∈ V
2 elmap.2 . 2 𝐵 ∈ V
3 fpmg 7769 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
41, 2, 3mp3an12 1406 1 (𝐹:𝐴𝐵𝐹 ∈ (𝐵pm 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173  ⟶wf 5800  (class class class)co 6549   ↑pm cpm 7745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747 This theorem is referenced by:  plycpn  23848  is1wlkg  40816  1wlkp1lem4  40885  0pthon-av  41295
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