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Theorem fmptdF 28267
Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6051 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 1805 . . . . 5  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  ->  [ y  /  x ] B  e.  C )
3 sban 2230 . . . . . 6  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] x  e.  A
) )
4 fmptdF.p . . . . . . . 8  |-  F/ x ph
54sbf 2211 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 fmptdF.a . . . . . . . 8  |-  F/_ x A
76clelsb3f 28128 . . . . . . 7  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
85, 7anbi12i 704 . . . . . 6  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] x  e.  A )  <->  ( ph  /\  y  e.  A ) )
93, 8bitri 253 . . . . 5  |-  ( [ y  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  y  e.  A ) )
10 sbsbc 3273 . . . . . 6  |-  ( [ y  /  x ] B  e.  C  <->  [. y  /  x ]. B  e.  C
)
11 sbcel12 3774 . . . . . . 7  |-  ( [. y  /  x ]. B  e.  C  <->  [_ y  /  x ]_ B  e.  [_ y  /  x ]_ C )
12 vex 3050 . . . . . . . . 9  |-  y  e. 
_V
13 fmptdF.c . . . . . . . . . 10  |-  F/_ x C
1413csbconstgf 3377 . . . . . . . . 9  |-  ( y  e.  _V  ->  [_ y  /  x ]_ C  =  C )
1512, 14ax-mp 5 . . . . . . . 8  |-  [_ y  /  x ]_ C  =  C
1615eleq2i 2523 . . . . . . 7  |-  ( [_ y  /  x ]_ B  e.  [_ y  /  x ]_ C  <->  [_ y  /  x ]_ B  e.  C
)
1711, 16bitri 253 . . . . . 6  |-  ( [. y  /  x ]. B  e.  C  <->  [_ y  /  x ]_ B  e.  C
)
1810, 17bitri 253 . . . . 5  |-  ( [ y  /  x ] B  e.  C  <->  [_ y  /  x ]_ B  e.  C
)
192, 9, 183imtr3i 269 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  C )
2019ralrimiva 2804 . . 3  |-  ( ph  ->  A. y  e.  A  [_ y  /  x ]_ B  e.  C )
21 nfcv 2594 . . . . 5  |-  F/_ y A
22 nfcv 2594 . . . . 5  |-  F/_ y B
23 nfcsb1v 3381 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
24 csbeq1a 3374 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
256, 21, 22, 23, 24cbvmptf 4496 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
2625fmpt 6048 . . 3  |-  ( A. y  e.  A  [_ y  /  x ]_ B  e.  C  <->  ( x  e.  A  |->  B ) : A --> C )
2720, 26sylib 200 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
28 fmptdF.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
2928feq1i 5725 . 2  |-  ( F : A --> C  <->  ( x  e.  A  |->  B ) : A --> C )
3027, 29sylibr 216 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446   F/wnf 1669   [wsb 1799    e. wcel 1889   F/_wnfc 2581   A.wral 2739   _Vcvv 3047   [.wsbc 3269   [_csb 3365    |-> cmpt 4464   -->wf 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593
This theorem is referenced by:  fmptcof2  28271  esumcl  28863  esumid  28877  esumgsum  28878  esumval  28879  esumel  28880  esumsplit  28886  esumaddf  28894  esumss  28905  esumpfinvalf  28909
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