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Theorem fmptdF 25994
Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5888 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 1706 . . . . 5  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  ->  [ y  /  x ] B  e.  C )
3 sban 2091 . . . . . 6  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] x  e.  A
) )
4 fmptdF.p . . . . . . . 8  |-  F/ x ph
54sbf 2071 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 fmptdF.a . . . . . . . 8  |-  F/_ x A
76clelsb3f 25886 . . . . . . 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 3211 . . . . . 6  |-  ( [ y  /  x ] B  e.  C  <->  [. y  /  x ]. B  e.  C
)
11 sbcel12 3696 . . . . . . 7  |-  ( [. y  /  x ]. B  e.  C  <->  [_ y  /  x ]_ B  e.  [_ y  /  x ]_ C )
12 vex 2996 . . . . . . . . 9  |-  y  e. 
_V
13 fmptdF.c . . . . . . . . . 10  |-  F/_ x C
1413csbconstgf 3321 . . . . . . . . 9  |-  ( y  e.  _V  ->  [_ y  /  x ]_ C  =  C )
1512, 14ax-mp 5 . . . . . . . 8  |-  [_ y  /  x ]_ C  =  C
1615eleq2i 2507 . . . . . . 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 2820 . . 3  |-  ( ph  ->  A. y  e.  A  [_ y  /  x ]_ B  e.  C )
21 nfcv 2589 . . . . 5  |-  F/_ y A
22 nfcv 2589 . . . . 5  |-  F/_ y B
23 nfcsb1v 3325 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
24 csbeq1a 3318 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
256, 21, 22, 23, 24cbvmptf 25993 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
2625fmpt 5885 . . 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 5572 . 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 1369   F/wnf 1589   [wsb 1700    e. wcel 1756   F/_wnfc 2575   A.wral 2736   _Vcvv 2993   [.wsbc 3207   [_csb 3309    e. cmpt 4371   -->wf 5435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447
This theorem is referenced by:  fmptcof2  26001  esumcl  26508  esumid  26521  esumval  26522  esumel  26523  esumsplit  26528  gsumesum  26532  esumlub  26533  esumaddf  26534  esumss  26543  esumpfinvalf  26547  voliune  26667
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