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Theorem ov2gf 6410
Description: The value of an operation class abstraction. A version of ovmpt2g 6420 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ov2gf.a  |-  F/_ x A
ov2gf.c  |-  F/_ y A
ov2gf.d  |-  F/_ y B
ov2gf.1  |-  F/_ x G
ov2gf.2  |-  F/_ y S
ov2gf.3  |-  ( x  =  A  ->  R  =  G )
ov2gf.4  |-  ( y  =  B  ->  G  =  S )
ov2gf.5  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ov2gf  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, C    x, D, y
Allowed substitution hints:    A( x, y)    B( x, y)    R( x, y)    S( x, y)    F( x, y)    G( x, y)    H( x, y)

Proof of Theorem ov2gf
StepHypRef Expression
1 elex 3070 . . 3  |-  ( S  e.  H  ->  S  e.  _V )
2 ov2gf.a . . . 4  |-  F/_ x A
3 ov2gf.c . . . 4  |-  F/_ y A
4 ov2gf.d . . . 4  |-  F/_ y B
5 ov2gf.1 . . . . . 6  |-  F/_ x G
65nfel1 2582 . . . . 5  |-  F/ x  G  e.  _V
7 ov2gf.5 . . . . . . . 8  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
8 nfmpt21 6347 . . . . . . . 8  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfcxfr 2564 . . . . . . 7  |-  F/_ x F
10 nfcv 2566 . . . . . . 7  |-  F/_ x
y
112, 9, 10nfov 6306 . . . . . 6  |-  F/_ x
( A F y )
1211, 5nfeq 2577 . . . . 5  |-  F/ x
( A F y )  =  G
136, 12nfim 1950 . . . 4  |-  F/ x
( G  e.  _V  ->  ( A F y )  =  G )
14 ov2gf.2 . . . . . 6  |-  F/_ y S
1514nfel1 2582 . . . . 5  |-  F/ y  S  e.  _V
16 nfmpt22 6348 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
177, 16nfcxfr 2564 . . . . . . 7  |-  F/_ y F
183, 17, 4nfov 6306 . . . . . 6  |-  F/_ y
( A F B )
1918, 14nfeq 2577 . . . . 5  |-  F/ y ( A F B )  =  S
2015, 19nfim 1950 . . . 4  |-  F/ y ( S  e.  _V  ->  ( A F B )  =  S )
21 ov2gf.3 . . . . . 6  |-  ( x  =  A  ->  R  =  G )
2221eleq1d 2473 . . . . 5  |-  ( x  =  A  ->  ( R  e.  _V  <->  G  e.  _V ) )
23 oveq1 6287 . . . . . 6  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
2423, 21eqeq12d 2426 . . . . 5  |-  ( x  =  A  ->  (
( x F y )  =  R  <->  ( A F y )  =  G ) )
2522, 24imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( R  e.  _V  ->  ( x F y )  =  R )  <-> 
( G  e.  _V  ->  ( A F y )  =  G ) ) )
26 ov2gf.4 . . . . . 6  |-  ( y  =  B  ->  G  =  S )
2726eleq1d 2473 . . . . 5  |-  ( y  =  B  ->  ( G  e.  _V  <->  S  e.  _V ) )
28 oveq2 6288 . . . . . 6  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
2928, 26eqeq12d 2426 . . . . 5  |-  ( y  =  B  ->  (
( A F y )  =  G  <->  ( A F B )  =  S ) )
3027, 29imbi12d 320 . . . 4  |-  ( y  =  B  ->  (
( G  e.  _V  ->  ( A F y )  =  G )  <-> 
( S  e.  _V  ->  ( A F B )  =  S ) ) )
317ovmpt4g 6408 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x F y )  =  R )
32313expia 1201 . . . 4  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( R  e.  _V  ->  ( x F y )  =  R ) )
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3126 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( S  e.  _V  ->  ( A F B )  =  S ) )
341, 33syl5 32 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( S  e.  H  ->  ( A F B )  =  S ) )
35343impia 1196 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   F/_wnfc 2552   _Vcvv 3061  (class class class)co 6280    |-> cmpt2 6282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285
This theorem is referenced by: (None)
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