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Theorem ovmpt2s 5971
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
ovmpt2s.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpt2s  |-  ( ( 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 ovmpt2s
StepHypRef Expression
1 elex 2796 . . 3  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V )
2 nfcv 2419 . . . . 5  |-  F/_ x A
3 nfcv 2419 . . . . 5  |-  F/_ y A
4 nfcv 2419 . . . . 5  |-  F/_ y B
5 nfcsb1v 3113 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ R
65nfel1 2429 . . . . . 6  |-  F/ x [_ A  /  x ]_ R  e.  _V
7 ovmpt2s.3 . . . . . . . . 9  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpt21 5914 . . . . . . . . 9  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2416 . . . . . . . 8  |-  F/_ x F
10 nfcv 2419 . . . . . . . 8  |-  F/_ x
y
112, 9, 10nfov 5881 . . . . . . 7  |-  F/_ x
( A F y )
1211, 5nfeq 2426 . . . . . 6  |-  F/ x
( A F y )  =  [_ A  /  x ]_ R
136, 12nfim 1769 . . . . 5  |-  F/ x
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )
14 nfcsb1v 3113 . . . . . . 7  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ R
1514nfel1 2429 . . . . . 6  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V
16 nfmpt22 5915 . . . . . . . . 9  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2416 . . . . . . . 8  |-  F/_ y F
183, 17, 4nfov 5881 . . . . . . 7  |-  F/_ y
( A F B )
1918, 14nfeq 2426 . . . . . 6  |-  F/ y ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R
2015, 19nfim 1769 . . . . 5  |-  F/ y ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
21 csbeq1a 3089 . . . . . . 7  |-  ( x  =  A  ->  R  =  [_ A  /  x ]_ R )
2221eleq1d 2349 . . . . . 6  |-  ( x  =  A  ->  ( R  e.  _V  <->  [_ A  /  x ]_ R  e.  _V ) )
23 oveq1 5865 . . . . . . 7  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2297 . . . . . 6  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  = 
[_ A  /  x ]_ R ) )
2522, 24imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R ) ) )
26 csbeq1a 3089 . . . . . . 7  |-  ( y  =  B  ->  [_ A  /  x ]_ R  = 
[_ B  /  y ]_ [_ A  /  x ]_ R )
2726eleq1d 2349 . . . . . 6  |-  ( y  =  B  ->  ( [_ A  /  x ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
28 oveq2 5866 . . . . . . 7  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2297 . . . . . 6  |-  ( y  =  B  ->  (
( A F y )  =  [_ A  /  x ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
3027, 29imbi12d 311 . . . . 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 5970 . . . . . 6  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1153 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2850 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
34 csbcomg 3104 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
3534eleq1d 2349 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
3634eqeq2d 2294 . . . 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 259 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
381, 37syl5 28 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
39383impia 1148 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081  (class class class)co 5858    e. cmpt2 5860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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