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Theorem fnmpt2ovd 6656
Description: A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.)
Hypotheses
Ref Expression
fnmpt2ovd.m  |-  ( ph  ->  M  Fn  ( A  X.  B ) )
fnmpt2ovd.s  |-  ( ( i  =  a  /\  j  =  b )  ->  D  =  C )
fnmpt2ovd.d  |-  ( (
ph  /\  i  e.  A  /\  j  e.  B
)  ->  D  e.  U )
fnmpt2ovd.c  |-  ( (
ph  /\  a  e.  A  /\  b  e.  B
)  ->  C  e.  V )
fnmpt2ovd.v  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y
) )
Assertion
Ref Expression
fnmpt2ovd  |-  ( ph  ->  ( M  =  ( a  e.  A , 
b  e.  B  |->  C )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  D ) )
Distinct variable groups:    A, a,
b, i, j    B, a, b, i, j    C, i, j    D, a, b    M, a, b, i, j    U, a, b, i, j    V, a, b, i, j    X, a, b, i, j    Y, a, b, i, j    ph, a, b, i, j
Allowed substitution hints:    C( a, b)    D( i, j)

Proof of Theorem fnmpt2ovd
StepHypRef Expression
1 fnmpt2ovd.m . . 3  |-  ( ph  ->  M  Fn  ( A  X.  B ) )
2 fnmpt2ovd.c . . . . . 6  |-  ( (
ph  /\  a  e.  A  /\  b  e.  B
)  ->  C  e.  V )
323expb 1188 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  B ) )  ->  C  e.  V )
43ralrimivva 2813 . . . 4  |-  ( ph  ->  A. a  e.  A  A. b  e.  B  C  e.  V )
5 eqid 2443 . . . . 5  |-  ( a  e.  A ,  b  e.  B  |->  C )  =  ( a  e.  A ,  b  e.  B  |->  C )
65fnmpt2 6647 . . . 4  |-  ( A. a  e.  A  A. b  e.  B  C  e.  V  ->  ( a  e.  A ,  b  e.  B  |->  C )  Fn  ( A  X.  B ) )
74, 6syl 16 . . 3  |-  ( ph  ->  ( a  e.  A ,  b  e.  B  |->  C )  Fn  ( A  X.  B ) )
8 eqfnov2 6202 . . 3  |-  ( ( M  Fn  ( A  X.  B )  /\  ( a  e.  A ,  b  e.  B  |->  C )  Fn  ( A  X.  B ) )  ->  ( M  =  ( a  e.  A ,  b  e.  B  |->  C )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  ( i ( a  e.  A ,  b  e.  B  |->  C ) j ) ) )
91, 7, 8syl2anc 661 . 2  |-  ( ph  ->  ( M  =  ( a  e.  A , 
b  e.  B  |->  C )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  ( i ( a  e.  A ,  b  e.  B  |->  C ) j ) ) )
10 nfcv 2584 . . . . . . . 8  |-  F/_ a D
11 nfcv 2584 . . . . . . . 8  |-  F/_ b D
12 nfcv 2584 . . . . . . . 8  |-  F/_ i C
13 nfcv 2584 . . . . . . . 8  |-  F/_ j C
14 fnmpt2ovd.s . . . . . . . 8  |-  ( ( i  =  a  /\  j  =  b )  ->  D  =  C )
1510, 11, 12, 13, 14cbvmpt2 6170 . . . . . . 7  |-  ( i  e.  A ,  j  e.  B  |->  D )  =  ( a  e.  A ,  b  e.  B  |->  C )
1615eqcomi 2447 . . . . . 6  |-  ( a  e.  A ,  b  e.  B  |->  C )  =  ( i  e.  A ,  j  e.  B  |->  D )
1716a1i 11 . . . . 5  |-  ( ph  ->  ( a  e.  A ,  b  e.  B  |->  C )  =  ( i  e.  A , 
j  e.  B  |->  D ) )
1817oveqd 6113 . . . 4  |-  ( ph  ->  ( i ( a  e.  A ,  b  e.  B  |->  C ) j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j ) )
1918eqeq2d 2454 . . 3  |-  ( ph  ->  ( ( i M j )  =  ( i ( a  e.  A ,  b  e.  B  |->  C ) j )  <->  ( i M j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j ) ) )
20192ralbidv 2762 . 2  |-  ( ph  ->  ( A. i  e.  A  A. j  e.  B  ( i M j )  =  ( i ( a  e.  A ,  b  e.  B  |->  C ) j )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j ) ) )
21 simprl 755 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
i  e.  A )
22 simprr 756 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
j  e.  B )
23 fnmpt2ovd.d . . . . . 6  |-  ( (
ph  /\  i  e.  A  /\  j  e.  B
)  ->  D  e.  U )
24233expb 1188 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  ->  D  e.  U )
25 eqid 2443 . . . . . 6  |-  ( i  e.  A ,  j  e.  B  |->  D )  =  ( i  e.  A ,  j  e.  B  |->  D )
2625ovmpt4g 6218 . . . . 5  |-  ( ( i  e.  A  /\  j  e.  B  /\  D  e.  U )  ->  ( i ( i  e.  A ,  j  e.  B  |->  D ) j )  =  D )
2721, 22, 24, 26syl3anc 1218 . . . 4  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
( i ( i  e.  A ,  j  e.  B  |->  D ) j )  =  D )
2827eqeq2d 2454 . . 3  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
( ( i M j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j )  <->  ( i M j )  =  D ) )
29282ralbidva 2760 . 2  |-  ( ph  ->  ( A. i  e.  A  A. j  e.  B  ( i M j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  D ) )
309, 20, 293bitrd 279 1  |-  ( ph  ->  ( M  =  ( a  e.  A , 
b  e.  B  |->  C )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720    X. cxp 4843    Fn wfn 5418  (class class class)co 6096    e. cmpt2 6098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583
This theorem is referenced by:  mpt2frlmd  18207
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