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Theorem ovmpt2dxf 5727
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 5630 . 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 2085 . . . . . . . . 9  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
87ovmpt4g 5724 . . . . . . . 8  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  _V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
98a1i 9 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D  /\  R  e. 
_V )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
106, 9alrimi 1458 . . . . . 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 2841 . . . . 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 1458 . . . 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 2841 . . 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 270 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  L )
15 simplr 497 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  =  A )
163ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  A  e.  C )
1715, 16eqeltrd 2161 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  e.  C )
185ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  B  e.  L )
19 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  =  B )
20 ovmpt2dx.3 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
2120adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  D  =  L )
2218, 19, 213eltr4d 2168 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  e.  D )
23 ovmpt2dx.2 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
2423anassrs 392 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  =  S )
25 ovmpt2dx.6 . . . . . . . . . 10  |-  ( ph  ->  S  e.  X )
26 elex 2624 . . . . . . . . . 10  |-  ( S  e.  X  ->  S  e.  _V )
2725, 26syl 14 . . . . . . . . 9  |-  ( ph  ->  S  e.  _V )
2827ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  S  e.  _V )
2924, 28eqeltrd 2161 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  e.  _V )
30 biimt 239 . . . . . . 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 1172 . . . . . 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 5631 . . . . . . 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 2099 . . . . . 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 188 . . . . 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 2236 . . . . . 6  |-  F/ y  x  =  A
376, 36nfan 1500 . . . . 5  |-  F/ y ( ph  /\  x  =  A )
38 nfmpt22 5673 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
39 nfcv 2225 . . . . . . . 8  |-  F/_ y B
4035, 38, 39nfov 5636 . . . . . . 7  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
41 ovmpt2dxf.sy . . . . . . 7  |-  F/_ y S
4240, 41nfeq 2232 . . . . . 6  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
4342a1i 9 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  F/ y ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  S )
4414, 34, 37, 43sbciedf 2863 . . . 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 2225 . . . . . . 7  |-  F/_ x A
46 nfmpt21 5672 . . . . . . 7  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
47 ovmpt2dxf.bx . . . . . . 7  |-  F/_ x B
4845, 46, 47nfov 5636 . . . . . 6  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
49 ovmpt2dxf.sx . . . . . 6  |-  F/_ x S
5048, 49nfeq 2232 . . . . 5  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
5150a1i 9 . . . 4  |-  ( ph  ->  F/ x ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
523, 44, 4, 51sbciedf 2863 . . 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 145 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
542, 53eqtrd 2117 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 922    = wceq 1287   F/wnf 1392    e. wcel 1436   F/_wnfc 2212   _Vcvv 2615   [.wsbc 2829  (class class class)co 5613    |-> cmpt2 5615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618
This theorem is referenced by:  ovmpt2dx  5728  mpt2xopoveq  5959
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