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Theorem ovmpt2s 6410
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 3122 . . 3  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V )
2 nfcv 2629 . . . . 5  |-  F/_ x A
3 nfcv 2629 . . . . 5  |-  F/_ y A
4 nfcv 2629 . . . . 5  |-  F/_ y B
5 nfcsb1v 3451 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ R
65nfel1 2645 . . . . . 6  |-  F/ x [_ A  /  x ]_ R  e.  _V
7 ovmpt2s.3 . . . . . . . . 9  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpt21 6348 . . . . . . . . 9  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2627 . . . . . . . 8  |-  F/_ x F
10 nfcv 2629 . . . . . . . 8  |-  F/_ x
y
112, 9, 10nfov 6307 . . . . . . 7  |-  F/_ x
( A F y )
1211, 5nfeq 2640 . . . . . 6  |-  F/ x
( A F y )  =  [_ A  /  x ]_ R
136, 12nfim 1867 . . . . 5  |-  F/ x
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )
14 nfcsb1v 3451 . . . . . . 7  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ R
1514nfel1 2645 . . . . . 6  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V
16 nfmpt22 6349 . . . . . . . . 9  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2627 . . . . . . . 8  |-  F/_ y F
183, 17, 4nfov 6307 . . . . . . 7  |-  F/_ y
( A F B )
1918, 14nfeq 2640 . . . . . 6  |-  F/ y ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R
2015, 19nfim 1867 . . . . 5  |-  F/ y ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
21 csbeq1a 3444 . . . . . . 7  |-  ( x  =  A  ->  R  =  [_ A  /  x ]_ R )
2221eleq1d 2536 . . . . . 6  |-  ( x  =  A  ->  ( R  e.  _V  <->  [_ A  /  x ]_ R  e.  _V ) )
23 oveq1 6291 . . . . . . 7  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2489 . . . . . 6  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  = 
[_ A  /  x ]_ R ) )
2522, 24imbi12d 320 . . . . 5  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R ) ) )
26 csbeq1a 3444 . . . . . . 7  |-  ( y  =  B  ->  [_ A  /  x ]_ R  = 
[_ B  /  y ]_ [_ A  /  x ]_ R )
2726eleq1d 2536 . . . . . 6  |-  ( y  =  B  ->  ( [_ A  /  x ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
28 oveq2 6292 . . . . . . 7  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2489 . . . . . 6  |-  ( y  =  B  ->  (
( A F y )  =  [_ A  /  x ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
3027, 29imbi12d 320 . . . . 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 6409 . . . . . 6  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1198 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3178 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
34 csbcom 3837 . . . . 5  |-  [_ A  /  x ]_ [_ B  /  y ]_ R  =  [_ B  /  y ]_ [_ A  /  x ]_ R
3534eleq1i 2544 . . . 4  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
3634eqeq2i 2485 . . . 4  |-  ( ( A F B )  =  [_ A  /  x ]_ [_ B  / 
y ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
3733, 35, 363imtr4g 270 . . 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 1193 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435  (class class class)co 6284    |-> cmpt2 6286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289
This theorem is referenced by: (None)
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