MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2dxf Structured version   Unicode version

Theorem ovmpt2dxf 6228
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 6127 . 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 2442 . . . . . . . . 9  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
87ovmpt4g 6225 . . . . . . . 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 1783 . . . . . 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 3180 . . . . 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 1783 . . . 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 3180 . . 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 453 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  L )
15 simplr 733 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  =  A )
163ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  A  e.  C )
1715, 16eqeltrd 2516 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  e.  C )
185ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  B  e.  L )
19 simpr 449 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  =  B )
20 ovmpt2dx.3 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
2120adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  D  =  L )
2218, 19, 213eltr4d 2523 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  e.  D )
23 ovmpt2dx.2 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
2423anassrs 631 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  =  S )
25 ovmpt2dx.6 . . . . . . . . . 10  |-  ( ph  ->  S  e.  X )
26 elex 2970 . . . . . . . . . 10  |-  ( S  e.  X  ->  S  e.  _V )
2725, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  S  e.  _V )
2827ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  S  e.  _V )
2924, 28eqeltrd 2516 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  e.  _V )
30 biimt 327 . . . . . . 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 1185 . . . . . 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 6128 . . . . . . 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 2456 . . . . . 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 248 . . . . 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 2589 . . . . . 6  |-  F/ y  x  =  A
376, 36nfan 1848 . . . . 5  |-  F/ y ( ph  /\  x  =  A )
38 nfmpt22 6170 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
39 nfcv 2578 . . . . . . . 8  |-  F/_ y B
4035, 38, 39nfov 6133 . . . . . . 7  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
41 ovmpt2dxf.sy . . . . . . 7  |-  F/_ y S
4240, 41nfeq 2585 . . . . . 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 3202 . . . 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 2578 . . . . . . 7  |-  F/_ x A
46 nfmpt21 6169 . . . . . . 7  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
47 ovmpt2dxf.bx . . . . . . 7  |-  F/_ x B
4845, 46, 47nfov 6133 . . . . . 6  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
49 ovmpt2dxf.sx . . . . . 6  |-  F/_ x S
5048, 49nfeq 2585 . . . . 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 3202 . . 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 203 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
542, 53eqtrd 2474 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   F/wnf 1554    = wceq 1653    e. wcel 1727   F/_wnfc 2565   _Vcvv 2962   [.wsbc 3167  (class class class)co 6110    e. cmpt2 6112
This theorem is referenced by:  ovmpt2dx  6229  mpt2xopoveq  6499  elovmpt2rab  28128  elovmpt2rab1  28129  ovmpt3rab1  28130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115
  Copyright terms: Public domain W3C validator