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Theorem cbvmptf 4083
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 1521 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 cbvmptf.1 . . . . . 6  |-  F/_ x A
32nfcri 2306 . . . . 5  |-  F/ x  w  e.  A
4 nfs1v 1932 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
53, 4nfan 1558 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
6 eleq1w 2231 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
7 sbequ12 1764 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
86, 7anbi12d 470 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
91, 5, 8cbvopab1 4062 . . 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 2306 . . . . 5  |-  F/ y  w  e.  A
12 cbvmptf.3 . . . . . . 7  |-  F/_ y B
1312nfeq2 2324 . . . . . 6  |-  F/ y  z  =  B
1413nfsb 1939 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
1511, 14nfan 1558 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
16 nfv 1521 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
17 eleq1w 2231 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
18 cbvmptf.4 . . . . . . 7  |-  F/_ x C
1918nfeq2 2324 . . . . . 6  |-  F/ x  z  =  C
20 cbvmptf.5 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
2120eqeq2d 2182 . . . . . 6  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2219, 21sbhypf 2779 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2317, 22anbi12d 470 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2415, 16, 23cbvopab1 4062 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
259, 24eqtri 2191 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 4052 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 4052 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2201 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348   [wsb 1755    e. wcel 2141   F/_wnfc 2299   {copab 4049    |-> cmpt 4050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-mpt 4052
This theorem is referenced by:  resmptf  4941
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