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Theorem cbvmptf 25983
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 1673 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 cbvmptf.1 . . . . . 6  |-  F/_ x A
32nfcri 2582 . . . . 5  |-  F/ x  w  e.  A
4 nfs1v 2142 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
53, 4nfan 1861 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
6 eleq1 2503 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
7 sbequ12 1936 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
86, 7anbi12d 710 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
91, 5, 8cbvopab1 4374 . . 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 2582 . . . . 5  |-  F/ y  w  e.  A
12 cbvmptf.3 . . . . . . 7  |-  F/_ y B
1312nfeq2 2605 . . . . . 6  |-  F/ y  z  =  B
1413nfsb 2146 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
1511, 14nfan 1861 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
16 nfv 1673 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
17 eleq1 2503 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
18 cbvmptf.4 . . . . . . 7  |-  F/_ x C
1918nfeq2 2605 . . . . . 6  |-  F/ x  z  =  C
20 cbvmptf.5 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
2120eqeq2d 2454 . . . . . 6  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2219, 21sbhypf 3031 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2317, 22anbi12d 710 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2415, 16, 23cbvopab1 4374 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
259, 24eqtri 2463 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 4364 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 4364 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2473 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   [wsb 1700    e. wcel 1756   F/_wnfc 2575   {copab 4361    e. cmpt 4362
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-opab 4363  df-mpt 4364
This theorem is referenced by:  fmptdF  25984  resmptf  25986  fvmpt2f  25987  offval2f  25995  funcnv4mpt  26001  cbvesum  26509  esumpfinvalf  26537
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