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Theorem ovmpt2dxf 6162
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 6061 . 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 2408 . . . . . . . . 9  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
87ovmpt4g 6159 . . . . . . . 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 1777 . . . . . 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 3138 . . . . 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 1777 . . . 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 3138 . . 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 452 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  L )
15 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  =  A )
163ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  A  e.  C )
1715, 16eqeltrd 2482 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  e.  C )
185ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  B  e.  L )
19 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  =  B )
20 ovmpt2dx.3 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
2120adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  D  =  L )
2218, 19, 213eltr4d 2489 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  e.  D )
23 ovmpt2dx.2 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
2423anassrs 630 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  =  S )
25 ovmpt2dx.6 . . . . . . . . . 10  |-  ( ph  ->  S  e.  X )
26 elex 2928 . . . . . . . . . 10  |-  ( S  e.  X  ->  S  e.  _V )
2725, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  S  e.  _V )
2827ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  S  e.  _V )
2924, 28eqeltrd 2482 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  e.  _V )
30 biimt 326 . . . . . . 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 1184 . . . . . 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 6062 . . . . . . 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 2422 . . . . . 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 247 . . . . 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 2555 . . . . . 6  |-  F/ y  x  =  A
376, 36nfan 1842 . . . . 5  |-  F/ y ( ph  /\  x  =  A )
38 nfmpt22 6104 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
39 nfcv 2544 . . . . . . . 8  |-  F/_ y B
4035, 38, 39nfov 6067 . . . . . . 7  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
41 ovmpt2dxf.sy . . . . . . 7  |-  F/_ y S
4240, 41nfeq 2551 . . . . . 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 3160 . . . 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 2544 . . . . . . 7  |-  F/_ x A
46 nfmpt21 6103 . . . . . . 7  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
47 ovmpt2dxf.bx . . . . . . 7  |-  F/_ x B
4845, 46, 47nfov 6067 . . . . . 6  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
49 ovmpt2dxf.sx . . . . . 6  |-  F/_ x S
5048, 49nfeq 2551 . . . . 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 3160 . . 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 202 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
542, 53eqtrd 2440 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2531   _Vcvv 2920   [.wsbc 3125  (class class class)co 6044    e. cmpt2 6046
This theorem is referenced by:  ovmpt2dx  6163  mpt2xopoveq  6433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049
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