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Theorem ovmpos 5860
Description: Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypothesis
Ref Expression
ovmpos.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpos  |-  ( ( A  e.  C  /\  B  e.  D  /\  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V )  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
)
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, D, y
Allowed substitution hints:    R( x, y)    F( x, y)    V( x, y)

Proof of Theorem ovmpos
StepHypRef Expression
1 elex 2669 . . 3  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V )
2 nfcv 2256 . . . . 5  |-  F/_ x A
3 nfcv 2256 . . . . 5  |-  F/_ y A
4 nfcv 2256 . . . . 5  |-  F/_ y B
5 nfcsb1v 3003 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ R
65nfel1 2267 . . . . . 6  |-  F/ x [_ A  /  x ]_ R  e.  _V
7 ovmpos.3 . . . . . . . . 9  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpo1 5804 . . . . . . . . 9  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2253 . . . . . . . 8  |-  F/_ x F
10 nfcv 2256 . . . . . . . 8  |-  F/_ x
y
112, 9, 10nfov 5767 . . . . . . 7  |-  F/_ x
( A F y )
1211, 5nfeq 2264 . . . . . 6  |-  F/ x
( A F y )  =  [_ A  /  x ]_ R
136, 12nfim 1534 . . . . 5  |-  F/ x
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )
14 nfcsb1v 3003 . . . . . . 7  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ R
1514nfel1 2267 . . . . . 6  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V
16 nfmpo2 5805 . . . . . . . . 9  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2253 . . . . . . . 8  |-  F/_ y F
183, 17, 4nfov 5767 . . . . . . 7  |-  F/_ y
( A F B )
1918, 14nfeq 2264 . . . . . 6  |-  F/ y ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R
2015, 19nfim 1534 . . . . 5  |-  F/ y ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
21 csbeq1a 2981 . . . . . . 7  |-  ( x  =  A  ->  R  =  [_ A  /  x ]_ R )
2221eleq1d 2184 . . . . . 6  |-  ( x  =  A  ->  ( R  e.  _V  <->  [_ A  /  x ]_ R  e.  _V ) )
23 oveq1 5747 . . . . . . 7  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2130 . . . . . 6  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  = 
[_ A  /  x ]_ R ) )
2522, 24imbi12d 233 . . . . 5  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R ) ) )
26 csbeq1a 2981 . . . . . . 7  |-  ( y  =  B  ->  [_ A  /  x ]_ R  = 
[_ B  /  y ]_ [_ A  /  x ]_ R )
2726eleq1d 2184 . . . . . 6  |-  ( y  =  B  ->  ( [_ A  /  x ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
28 oveq2 5748 . . . . . . 7  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2130 . . . . . 6  |-  ( y  =  B  ->  (
( A F y )  =  [_ A  /  x ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
3027, 29imbi12d 233 . . . . 5  |-  ( y  =  B  ->  (
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )  <-> 
( [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) ) )
317ovmpt4g 5859 . . . . . 6  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1166 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2725 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
34 csbcomg 2994 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
3534eleq1d 2184 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
3634eqeq2d 2127 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R  <->  ( A F B )  =  [_ B  / 
y ]_ [_ A  /  x ]_ R ) )
3733, 35, 363imtr4d 202 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
381, 37syl5 32 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
39383impia 1161 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V )  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   _Vcvv 2658   [_csb 2973  (class class class)co 5740    e. cmpo 5742
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745
This theorem is referenced by: (None)
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