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Theorem ovmpt2dxf 6409
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
ovmpt2dx.2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
ovmpt2dx.3  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
ovmpt2dx.4  |-  ( ph  ->  A  e.  C )
ovmpt2dx.5  |-  ( ph  ->  B  e.  L )
ovmpt2dx.6  |-  ( ph  ->  S  e.  X )
ovmpt2dxf.px  |-  F/ x ph
ovmpt2dxf.py  |-  F/ y
ph
ovmpt2dxf.ay  |-  F/_ y A
ovmpt2dxf.bx  |-  F/_ x B
ovmpt2dxf.sx  |-  F/_ x S
ovmpt2dxf.sy  |-  F/_ y S
Assertion
Ref Expression
ovmpt2dxf  |-  ( ph  ->  ( A F B )  =  S )
Distinct variable groups:    x, y    x, A    y, B
Allowed substitution hints:    ph( x, y)    A( y)    B( x)    C( x, y)    D( x, y)    R( x, y)    S( x, y)    F( x, y)    L( x, y)    X( x, y)

Proof of Theorem ovmpt2dxf
StepHypRef Expression
1 ovmpt2dx.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
21oveqd 6295 . 2  |-  ( ph  ->  ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
3 ovmpt2dx.4 . . . 4  |-  ( ph  ->  A  e.  C )
4 ovmpt2dxf.px . . . . 5  |-  F/ x ph
5 ovmpt2dx.5 . . . . . 6  |-  ( ph  ->  B  e.  L )
6 ovmpt2dxf.py . . . . . . 7  |-  F/ y
ph
7 eqid 2402 . . . . . . . . 9  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
87ovmpt4g 6406 . . . . . . . 8  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
98a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
106, 9alrimi 1901 . . . . . 6  |-  ( ph  ->  A. y ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
115, 10spsbcd 3291 . . . . 5  |-  ( ph  ->  [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
124, 11alrimi 1901 . . . 4  |-  ( ph  ->  A. x [. B  /  y ]. (
( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x ( x  e.  C , 
y  e.  D  |->  R ) y )  =  R ) )
133, 12spsbcd 3291 . . 3  |-  ( ph  ->  [. A  /  x ]. [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
145adantr 463 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  L )
15 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  =  A )
163ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  A  e.  C )
1715, 16eqeltrd 2490 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  e.  C )
185ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  B  e.  L )
19 simpr 459 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  =  B )
20 ovmpt2dx.3 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
2120adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  D  =  L )
2218, 19, 213eltr4d 2505 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  e.  D )
23 ovmpt2dx.2 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
2423anassrs 646 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  =  S )
25 ovmpt2dx.6 . . . . . . . . . 10  |-  ( ph  ->  S  e.  X )
26 elex 3068 . . . . . . . . . 10  |-  ( S  e.  X  ->  S  e.  _V )
2725, 26syl 17 . . . . . . . . 9  |-  ( ph  ->  S  e.  _V )
2827ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  S  e.  _V )
2924, 28eqeltrd 2490 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  e.  _V )
30 biimt 333 . . . . . . 7  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( ( x ( x  e.  C , 
y  e.  D  |->  R ) y )  =  R  <->  ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) ) )
3117, 22, 29, 30syl3anc 1230 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R  <-> 
( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) ) )
3215, 19oveq12d 6296 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
3332, 24eqeq12d 2424 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
3431, 33bitr3d 255 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
35 ovmpt2dxf.ay . . . . . . 7  |-  F/_ y A
3635nfeq2 2581 . . . . . 6  |-  F/ y  x  =  A
376, 36nfan 1956 . . . . 5  |-  F/ y ( ph  /\  x  =  A )
38 nfmpt22 6346 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
39 nfcv 2564 . . . . . . . 8  |-  F/_ y B
4035, 38, 39nfov 6304 . . . . . . 7  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
41 ovmpt2dxf.sy . . . . . . 7  |-  F/_ y S
4240, 41nfeq 2575 . . . . . 6  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
4342a1i 11 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  F/ y ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  S )
4414, 34, 37, 43sbciedf 3313 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
45 nfcv 2564 . . . . . . 7  |-  F/_ x A
46 nfmpt21 6345 . . . . . . 7  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
47 ovmpt2dxf.bx . . . . . . 7  |-  F/_ x B
4845, 46, 47nfov 6304 . . . . . 6  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
49 ovmpt2dxf.sx . . . . . 6  |-  F/_ x S
5048, 49nfeq 2575 . . . . 5  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
5150a1i 11 . . . 4  |-  ( ph  ->  F/ x ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
523, 44, 4, 51sbciedf 3313 . . 3  |-  ( ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
5313, 52mpbid 210 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
542, 53eqtrd 2443 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   F/wnf 1637    e. wcel 1842   F/_wnfc 2550   _Vcvv 3059   [.wsbc 3277  (class class class)co 6278    |-> cmpt2 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283
This theorem is referenced by:  ovmpt2dx  6410  elovmpt2rab  6502  elovmpt2rab1  6503  ovmpt3rab1  6515  mpt2xopoveq  6950  fvmpt2curryd  7003  mdetralt2  19403  mdetunilem2  19407  gsummatr01lem4  19452
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