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Theorem fpmg 8631
Description: A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
Assertion
Ref Expression
fpmg ((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))

Proof of Theorem fpmg
StepHypRef Expression
1 ssid 3948 . . . 4 𝐴𝐴
2 elpm2r 8608 . . . 4 (((𝐵𝑊𝐴𝑉) ∧ (𝐹:𝐴𝐵𝐴𝐴)) → 𝐹 ∈ (𝐵pm 𝐴))
31, 2mpanr2 701 . . 3 (((𝐵𝑊𝐴𝑉) ∧ 𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
433impa 1109 . 2 ((𝐵𝑊𝐴𝑉𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
543com12 1122 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wcel 2110  wss 3892  wf 6427  (class class class)co 7269  pm cpm 8591
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 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-pm 8593
This theorem is referenced by:  fpm  8638  mapsspm  8639  dvnff  25077  dvnply2  25437  0wlkonlem2  28471
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