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Theorem fmptdF 27318
Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6056 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
Hypotheses
Ref Expression
fmptdF.p  |-  F/ x ph
fmptdF.a  |-  F/_ x A
fmptdF.c  |-  F/_ x C
fmptdF.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
fmptdF.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fmptdF  |-  ( ph  ->  F : A --> C )

Proof of Theorem fmptdF
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fmptdF.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
21sbimi 1717 . . . . 5  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  ->  [ y  /  x ] B  e.  C )
3 sban 2114 . . . . . 6  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] x  e.  A
) )
4 fmptdF.p . . . . . . . 8  |-  F/ x ph
54sbf 2094 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 fmptdF.a . . . . . . . 8  |-  F/_ x A
76clelsb3f 27202 . . . . . . 7  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
85, 7anbi12i 697 . . . . . 6  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] x  e.  A )  <->  ( ph  /\  y  e.  A ) )
93, 8bitri 249 . . . . 5  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  y  e.  A ) )
10 sbsbc 3340 . . . . . 6  |-  ( [ y  /  x ] B  e.  C  <->  [. y  /  x ]. B  e.  C
)
11 sbcel12 3828 . . . . . . 7  |-  ( [. y  /  x ]. B  e.  C  <->  [_ y  /  x ]_ B  e.  [_ y  /  x ]_ C )
12 vex 3121 . . . . . . . . 9  |-  y  e. 
_V
13 fmptdF.c . . . . . . . . . 10  |-  F/_ x C
1413csbconstgf 3452 . . . . . . . . 9  |-  ( y  e.  _V  ->  [_ y  /  x ]_ C  =  C )
1512, 14ax-mp 5 . . . . . . . 8  |-  [_ y  /  x ]_ C  =  C
1615eleq2i 2545 . . . . . . 7  |-  ( [_ y  /  x ]_ B  e.  [_ y  /  x ]_ C  <->  [_ y  /  x ]_ B  e.  C
)
1711, 16bitri 249 . . . . . 6  |-  ( [. y  /  x ]. B  e.  C  <->  [_ y  /  x ]_ B  e.  C
)
1810, 17bitri 249 . . . . 5  |-  ( [ y  /  x ] B  e.  C  <->  [_ y  /  x ]_ B  e.  C
)
192, 9, 183imtr3i 265 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  C )
2019ralrimiva 2881 . . 3  |-  ( ph  ->  A. y  e.  A  [_ y  /  x ]_ B  e.  C )
21 nfcv 2629 . . . . 5  |-  F/_ y A
22 nfcv 2629 . . . . 5  |-  F/_ y B
23 nfcsb1v 3456 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
24 csbeq1a 3449 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
256, 21, 22, 23, 24cbvmptf 27317 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
2625fmpt 6053 . . 3  |-  ( A. y  e.  A  [_ y  /  x ]_ B  e.  C  <->  ( x  e.  A  |->  B ) : A --> C )
2720, 26sylib 196 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
28 fmptdF.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
2928feq1i 5729 . 2  |-  ( F : A --> C  <->  ( x  e.  A  |->  B ) : A --> C )
3027, 29sylibr 212 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   F/wnf 1599   [wsb 1711    e. wcel 1767   F/_wnfc 2615   A.wral 2817   _Vcvv 3118   [.wsbc 3336   [_csb 3440    |-> cmpt 4511   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  fmptcof2  27325  esumcl  27868  esumid  27881  esumval  27882  esumel  27883  esumsplit  27888  esumaddf  27894  esumss  27903  esumpfinvalf  27907  voliune  28026
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