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Theorem fmpt2d 5890
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 2781 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2435 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5563 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 16 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 fmpt2d.1 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5528 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 224 . 2  |-  ( ph  ->  F  Fn  A )
9 fmpt2d.3 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
109ralrimiva 2781 . 2  |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  C )
11 ffnfv 5886 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  e.  C
) )
128, 10, 11sylanbrc 646 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258    Fn wfn 5441   -->wf 5442   ` cfv 5446
This theorem is referenced by:  fmpt2dOLD  5891  cantnff  7621  limsupgre  12267  idaf  14210  curfcl  14321  yonedainv  14370  clsf  17104  kgenf  17565  vmaf  20894  lgsdchr  21124  lgamf  24818  erdszelem6  24874  cdleme50f  31276  dochfN  32091
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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