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Theorem fmpt2dOLD 5689
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 9-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fmpt2dOLD.1  |-  ( ph  ->  ( x  e.  A  ->  B  e.  V ) )
fmpt2dOLD.2  |-  F  =  ( x  e.  A  |->  B )
fmpt2dOLD.3  |-  ( ph  ->  ( y  e.  A  ->  ( F `  y
)  e.  C ) )
Assertion
Ref Expression
fmpt2dOLD  |-  ( ph  ->  F : A --> C )
Distinct variable groups:    x, y, A    y, C    y, F    ph, x, y
Allowed substitution hints:    B( x, y)    C( x)    F( x)    V( x, y)

Proof of Theorem fmpt2dOLD
StepHypRef Expression
1 fmpt2dOLD.1 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  e.  V ) )
21imp 418 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
3 fmpt2dOLD.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
43a1i 10 . 2  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
5 fmpt2dOLD.3 . . 3  |-  ( ph  ->  ( y  e.  A  ->  ( F `  y
)  e.  C ) )
65imp 418 . 2  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
72, 4, 6fmpt2d 5688 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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