MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2s Structured version   Unicode version

Theorem ovmpt2s 6200
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 2966 . . 3  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V )
2 nfcv 2574 . . . . 5  |-  F/_ x A
3 nfcv 2574 . . . . 5  |-  F/_ y A
4 nfcv 2574 . . . . 5  |-  F/_ y B
5 nfcsb1v 3285 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ R
65nfel1 2584 . . . . . 6  |-  F/ x [_ A  /  x ]_ R  e.  _V
7 ovmpt2s.3 . . . . . . . . 9  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpt21 6143 . . . . . . . . 9  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2571 . . . . . . . 8  |-  F/_ x F
10 nfcv 2574 . . . . . . . 8  |-  F/_ x
y
112, 9, 10nfov 6107 . . . . . . 7  |-  F/_ x
( A F y )
1211, 5nfeq 2581 . . . . . 6  |-  F/ x
( A F y )  =  [_ A  /  x ]_ R
136, 12nfim 1833 . . . . 5  |-  F/ x
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )
14 nfcsb1v 3285 . . . . . . 7  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ R
1514nfel1 2584 . . . . . 6  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V
16 nfmpt22 6144 . . . . . . . . 9  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2571 . . . . . . . 8  |-  F/_ y F
183, 17, 4nfov 6107 . . . . . . 7  |-  F/_ y
( A F B )
1918, 14nfeq 2581 . . . . . 6  |-  F/ y ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R
2015, 19nfim 1833 . . . . 5  |-  F/ y ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
21 csbeq1a 3261 . . . . . . 7  |-  ( x  =  A  ->  R  =  [_ A  /  x ]_ R )
2221eleq1d 2504 . . . . . 6  |-  ( x  =  A  ->  ( R  e.  _V  <->  [_ A  /  x ]_ R  e.  _V ) )
23 oveq1 6091 . . . . . . 7  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2452 . . . . . 6  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  = 
[_ A  /  x ]_ R ) )
2522, 24imbi12d 313 . . . . 5  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R ) ) )
26 csbeq1a 3261 . . . . . . 7  |-  ( y  =  B  ->  [_ A  /  x ]_ R  = 
[_ B  /  y ]_ [_ A  /  x ]_ R )
2726eleq1d 2504 . . . . . 6  |-  ( y  =  B  ->  ( [_ A  /  x ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
28 oveq2 6092 . . . . . . 7  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2452 . . . . . 6  |-  ( y  =  B  ->  (
( A F y )  =  [_ A  /  x ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
3027, 29imbi12d 313 . . . . 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 6199 . . . . . 6  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1156 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3020 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
34 csbcomg 3276 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
3534eleq1d 2504 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
3634eqeq2d 2449 . . . 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 261 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
381, 37syl5 31 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
39383impia 1151 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   [_csb 3253  (class class class)co 6084    e. cmpt2 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
  Copyright terms: Public domain W3C validator