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Theorem ovmpt2s 6425
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 3087 . . 3  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V )
2 nfcv 2582 . . . . 5  |-  F/_ x A
3 nfcv 2582 . . . . 5  |-  F/_ y A
4 nfcv 2582 . . . . 5  |-  F/_ y B
5 nfcsb1v 3408 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ R
65nfel1 2598 . . . . . 6  |-  F/ x [_ A  /  x ]_ R  e.  _V
7 ovmpt2s.3 . . . . . . . . 9  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpt21 6363 . . . . . . . . 9  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2580 . . . . . . . 8  |-  F/_ x F
10 nfcv 2582 . . . . . . . 8  |-  F/_ x
y
112, 9, 10nfov 6322 . . . . . . 7  |-  F/_ x
( A F y )
1211, 5nfeq 2593 . . . . . 6  |-  F/ x
( A F y )  =  [_ A  /  x ]_ R
136, 12nfim 1975 . . . . 5  |-  F/ x
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R )
14 nfcsb1v 3408 . . . . . . 7  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ R
1514nfel1 2598 . . . . . 6  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V
16 nfmpt22 6364 . . . . . . . . 9  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2580 . . . . . . . 8  |-  F/_ y F
183, 17, 4nfov 6322 . . . . . . 7  |-  F/_ y
( A F B )
1918, 14nfeq 2593 . . . . . 6  |-  F/ y ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R
2015, 19nfim 1975 . . . . 5  |-  F/ y ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
21 csbeq1a 3401 . . . . . . 7  |-  ( x  =  A  ->  R  =  [_ A  /  x ]_ R )
2221eleq1d 2489 . . . . . 6  |-  ( x  =  A  ->  ( R  e.  _V  <->  [_ A  /  x ]_ R  e.  _V ) )
23 oveq1 6303 . . . . . . 7  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2442 . . . . . 6  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  = 
[_ A  /  x ]_ R ) )
2522, 24imbi12d 321 . . . . 5  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( [_ A  /  x ]_ R  e.  _V  ->  ( A F y )  =  [_ A  /  x ]_ R ) ) )
26 csbeq1a 3401 . . . . . . 7  |-  ( y  =  B  ->  [_ A  /  x ]_ R  = 
[_ B  /  y ]_ [_ A  /  x ]_ R )
2726eleq1d 2489 . . . . . 6  |-  ( y  =  B  ->  ( [_ A  /  x ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
)
28 oveq2 6304 . . . . . . 7  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2442 . . . . . 6  |-  ( y  =  B  ->  (
( A F y )  =  [_ A  /  x ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
3027, 29imbi12d 321 . . . . 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 6424 . . . . . 6  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1207 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3143 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ B  / 
y ]_ [_ A  /  x ]_ R  e.  _V  ->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R ) )
34 csbcom 3808 . . . . 5  |-  [_ A  /  x ]_ [_ B  /  y ]_ R  =  [_ B  /  y ]_ [_ A  /  x ]_ R
3534eleq1i 2497 . . . 4  |-  ( [_ A  /  x ]_ [_ B  /  y ]_ R  e.  _V  <->  [_ B  /  y ]_ [_ A  /  x ]_ R  e.  _V )
3634eqeq2i 2438 . . . 4  |-  ( ( A F B )  =  [_ A  /  x ]_ [_ B  / 
y ]_ R  <->  ( A F B )  =  [_ B  /  y ]_ [_ A  /  x ]_ R )
3733, 35, 363imtr4g 273 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  _V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
381, 37syl5 33 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( [_ A  /  x ]_ [_ B  / 
y ]_ R  e.  V  ->  ( A F B )  =  [_ A  /  x ]_ [_ B  /  y ]_ R
) )
39383impia 1202 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078   [_csb 3392  (class class class)co 6296    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by: (None)
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