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Theorem ovmpt2rdxf 33182
Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6401. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
ovmpt2rdx.1  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
ovmpt2rdx.2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
ovmpt2rdx.3  |-  ( (
ph  /\  y  =  B )  ->  C  =  L )
ovmpt2rdx.4  |-  ( ph  ->  A  e.  L )
ovmpt2rdx.5  |-  ( ph  ->  B  e.  D )
ovmpt2rdx.6  |-  ( ph  ->  S  e.  X )
ovmpt2rdxf.px  |-  F/ x ph
ovmpt2rdxf.py  |-  F/ y
ph
ovmpt2rdxf.ay  |-  F/_ y A
ovmpt2rdxf.bx  |-  F/_ x B
ovmpt2rdxf.sx  |-  F/_ x S
ovmpt2rdxf.sy  |-  F/_ y S
Assertion
Ref Expression
ovmpt2rdxf  |-  ( 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 ovmpt2rdxf
StepHypRef Expression
1 ovmpt2rdx.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
21oveqd 6287 . 2  |-  ( ph  ->  ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
3 ovmpt2rdx.4 . . . 4  |-  ( ph  ->  A  e.  L )
4 ovmpt2rdxf.px . . . . 5  |-  F/ x ph
5 ovmpt2rdx.5 . . . . . 6  |-  ( ph  ->  B  e.  D )
6 ovmpt2rdxf.py . . . . . . 7  |-  F/ y
ph
7 eqid 2454 . . . . . . . . 9  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
87ovmpt4g 6398 . . . . . . . 8  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
98a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
106, 9alrimi 1882 . . . . . 6  |-  ( ph  ->  A. y ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
115, 10spsbcd 3338 . . . . 5  |-  ( ph  ->  [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
124, 11alrimi 1882 . . . 4  |-  ( ph  ->  A. x [. B  /  y ]. (
( x  e.  C  /\  y  e.  D  /\  R  e.  X
)  ->  ( x
( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
133, 12spsbcd 3338 . . 3  |-  ( ph  ->  [. A  /  x ]. [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) )
145adantr 463 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
153ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  A  e.  L )
16 simpr 459 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1716adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  =  A )
18 ovmpt2rdx.3 . . . . . . . . 9  |-  ( (
ph  /\  y  =  B )  ->  C  =  L )
1918adantlr 712 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  C  =  L )
2015, 17, 193eltr4d 2557 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  x  e.  C )
215ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  B  e.  D )
22 eleq1 2526 . . . . . . . . 9  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
2322adantl 464 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
y  e.  D  <->  B  e.  D ) )
2421, 23mpbird 232 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  e.  D )
25 ovmpt2rdx.2 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
2625anassrs 646 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  =  S )
27 ovmpt2rdx.6 . . . . . . . . 9  |-  ( ph  ->  S  e.  X )
2827ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  S  e.  X )
2926, 28eqeltrd 2542 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  R  e.  X )
30 biimt 333 . . . . . . 7  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  ( ( x ( x  e.  C , 
y  e.  D  |->  R ) y )  =  R  <->  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) ) )
3120, 24, 29, 30syl3anc 1226 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R  <-> 
( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R ) ) )
32 simpr 459 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  y  =  B )
3317, 32oveq12d 6288 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
3433, 26eqeq12d 2476 . . . . . 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 ) )
3531, 34bitr3d 255 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  (
( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
36 ovmpt2rdxf.ay . . . . . . 7  |-  F/_ y A
3736nfeq2 2633 . . . . . 6  |-  F/ y  x  =  A
386, 37nfan 1933 . . . . 5  |-  F/ y ( ph  /\  x  =  A )
39 nfmpt22 6338 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
40 nfcv 2616 . . . . . . . 8  |-  F/_ y B
4136, 39, 40nfov 6296 . . . . . . 7  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
42 ovmpt2rdxf.sy . . . . . . 7  |-  F/_ y S
4341, 42nfeq 2627 . . . . . 6  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
4443a1i 11 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  F/ y ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  S )
4514, 35, 38, 44sbciedf 3360 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( [. B  /  y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
46 nfcv 2616 . . . . . . 7  |-  F/_ x A
47 nfmpt21 6337 . . . . . . 7  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
48 ovmpt2rdxf.bx . . . . . . 7  |-  F/_ x B
4946, 47, 48nfov 6296 . . . . . 6  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
50 ovmpt2rdxf.sx . . . . . 6  |-  F/_ x S
5149, 50nfeq 2627 . . . . 5  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
5251a1i 11 . . . 4  |-  ( ph  ->  F/ x ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
533, 45, 4, 52sbciedf 3360 . . 3  |-  ( ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ( ( x  e.  C  /\  y  e.  D  /\  R  e.  X )  ->  (
x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
5413, 53mpbid 210 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
552, 54eqtrd 2495 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   F/wnf 1621    e. wcel 1823   F/_wnfc 2602   [.wsbc 3324  (class class class)co 6270    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  ovmpt2rdx  33183
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