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Theorem fpmg 6809
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 3210 . . . 4  |-  A  C_  A
2 elpm2r 6804 . . . 4  |-  ( ( ( B  e.  W  /\  A  e.  V
)  /\  ( F : A --> B  /\  A  C_  A ) )  ->  F  e.  ( B  ^pm  A ) )
31, 2mpanr2 665 . . 3  |-  ( ( ( B  e.  W  /\  A  e.  V
)  /\  F : A
--> B )  ->  F  e.  ( B  ^pm  A
) )
433impa 1146 . 2  |-  ( ( B  e.  W  /\  A  e.  V  /\  F : A --> B )  ->  F  e.  ( B  ^pm  A )
)
543com12 1155 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 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   -->wf 5267  (class class class)co 5874    ^pm cpm 6789
This theorem is referenced by:  fpm  6816  mapsspm  6817  dvnff  19288  dvnply2  19683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791
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