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Theorem fmpt2d 5550
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmpt2d.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
fmpt2d.1  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
fmpt2d.3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
Assertion
Ref Expression
fmpt2d  |-  ( ph  ->  F : A --> C )
Distinct variable groups:    x, A    y, A    y, C    y, F    ph, x    ph, y
Allowed substitution hints:    B( x, y)    C( x)    F( x)    V( x, y)

Proof of Theorem fmpt2d
StepHypRef Expression
1 fmpt2d.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
21ralrimiva 2482 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2117 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5219 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 14 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 fmpt2d.1 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5183 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 166 . 2  |-  ( ph  ->  F  Fn  A )
9 fmpt2d.3 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
109ralrimiva 2482 . 2  |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  C )
11 ffnfv 5546 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  e.  C
) )
128, 10, 11sylanbrc 413 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393    |-> cmpt 3959    Fn wfn 5088   -->wf 5089   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by: (None)
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