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Theorem fpmg 7032
Description: A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
Assertion
Ref Expression
fpmg  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A --> B )  ->  F  e.  ( B  ^pm  A )
)

Proof of Theorem fpmg
StepHypRef Expression
1 ssid 3360 . . . 4  |-  A  C_  A
2 elpm2r 7027 . . . 4  |-  ( ( ( B  e.  W  /\  A  e.  V
)  /\  ( F : A --> B  /\  A  C_  A ) )  ->  F  e.  ( B  ^pm  A ) )
31, 2mpanr2 666 . . 3  |-  ( ( ( B  e.  W  /\  A  e.  V
)  /\  F : A
--> B )  ->  F  e.  ( B  ^pm  A
) )
433impa 1148 . 2  |-  ( ( B  e.  W  /\  A  e.  V  /\  F : A --> B )  ->  F  e.  ( B  ^pm  A )
)
543com12 1157 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A --> B )  ->  F  e.  ( B  ^pm  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    C_ wss 3313   -->wf 5443  (class class class)co 6074    ^pm cpm 7012
This theorem is referenced by:  fpm  7039  mapsspm  7040  dvnff  19802  dvnply2  20197  0wlkon  21540  0trlon  21541  0pthon  21572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-pm 7014
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