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Theorem fmpt2dOLD 5901
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 420 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
3 fmpt2dOLD.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
43a1i 11 . 2  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
5 fmpt2dOLD.3 . . 3  |-  ( ph  ->  ( y  e.  A  ->  ( F `  y
)  e.  C ) )
65imp 420 . 2  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
72, 4, 6fmpt2d 5900 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4268   -->wf 5452   ` cfv 5456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464
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