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Theorem cbvmptf 27718
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
Hypotheses
Ref Expression
cbvmptf.1  |-  F/_ x A
cbvmptf.2  |-  F/_ y A
cbvmptf.3  |-  F/_ y B
cbvmptf.4  |-  F/_ x C
cbvmptf.5  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvmptf  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem cbvmptf
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1712 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 cbvmptf.1 . . . . . 6  |-  F/_ x A
32nfcri 2609 . . . . 5  |-  F/ x  w  e.  A
4 nfs1v 2183 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
53, 4nfan 1933 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
6 eleq1 2526 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
7 sbequ12 1997 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
86, 7anbi12d 708 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
91, 5, 8cbvopab1 4509 . . 3  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }
10 cbvmptf.2 . . . . . 6  |-  F/_ y A
1110nfcri 2609 . . . . 5  |-  F/ y  w  e.  A
12 cbvmptf.3 . . . . . . 7  |-  F/_ y B
1312nfeq2 2633 . . . . . 6  |-  F/ y  z  =  B
1413nfsb 2186 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
1511, 14nfan 1933 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
16 nfv 1712 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
17 eleq1 2526 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
18 cbvmptf.4 . . . . . . 7  |-  F/_ x C
1918nfeq2 2633 . . . . . 6  |-  F/ x  z  =  C
20 cbvmptf.5 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
2120eqeq2d 2468 . . . . . 6  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2219, 21sbhypf 3153 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2317, 22anbi12d 708 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2415, 16, 23cbvopab1 4509 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
259, 24eqtri 2483 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 4499 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 4499 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2493 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   [wsb 1744    e. wcel 1823   F/_wnfc 2602   {copab 4496    |-> cmpt 4497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-mpt 4499
This theorem is referenced by:  fmptdF  27719  resmptf  27722  fvmpt2f  27723  acunirnmpt2f  27731  offval2f  27736  funcnv4mpt  27742  cbvesum  28274  esumpfinvalf  28308  binomcxplemdvbinom  31502  binomcxplemdvsum  31504  binomcxplemnotnn0  31505
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