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Theorem fnmpoovd 6191
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.) (Revised by AV, 3-Jul-2022.)
Hypotheses
Ref Expression
fnmpoovd.m  |-  ( ph  ->  M  Fn  ( A  X.  B ) )
fnmpoovd.s  |-  ( ( i  =  a  /\  j  =  b )  ->  D  =  C )
fnmpoovd.d  |-  ( (
ph  /\  i  e.  A  /\  j  e.  B
)  ->  D  e.  U )
fnmpoovd.c  |-  ( (
ph  /\  a  e.  A  /\  b  e.  B
)  ->  C  e.  V )
Assertion
Ref Expression
fnmpoovd  |-  ( 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   
i, M, j    ph, a,
b, i, j
Allowed substitution hints:    C( a, b)    D( i, j)    U( i, j, a, b)    M( a, b)    V( i, j, a, b)

Proof of Theorem fnmpoovd
StepHypRef Expression
1 fnmpoovd.m . . 3  |-  ( ph  ->  M  Fn  ( A  X.  B ) )
2 fnmpoovd.c . . . . . 6  |-  ( (
ph  /\  a  e.  A  /\  b  e.  B
)  ->  C  e.  V )
323expb 1199 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  B ) )  ->  C  e.  V )
43ralrimivva 2552 . . . 4  |-  ( ph  ->  A. a  e.  A  A. b  e.  B  C  e.  V )
5 eqid 2170 . . . . 5  |-  ( a  e.  A ,  b  e.  B  |->  C )  =  ( a  e.  A ,  b  e.  B  |->  C )
65fnmpo 6178 . . . 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 14 . . 3  |-  ( ph  ->  ( a  e.  A ,  b  e.  B  |->  C )  Fn  ( A  X.  B ) )
8 eqfnov2 5957 . . 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 409 . 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 2312 . . . . . . . 8  |-  F/_ a D
11 nfcv 2312 . . . . . . . 8  |-  F/_ b D
12 nfcv 2312 . . . . . . . 8  |-  F/_ i C
13 nfcv 2312 . . . . . . . 8  |-  F/_ j C
14 fnmpoovd.s . . . . . . . 8  |-  ( ( i  =  a  /\  j  =  b )  ->  D  =  C )
1510, 11, 12, 13, 14cbvmpo 5929 . . . . . . 7  |-  ( i  e.  A ,  j  e.  B  |->  D )  =  ( a  e.  A ,  b  e.  B  |->  C )
1615eqcomi 2174 . . . . . 6  |-  ( a  e.  A ,  b  e.  B  |->  C )  =  ( i  e.  A ,  j  e.  B  |->  D )
1716a1i 9 . . . . 5  |-  ( ph  ->  ( a  e.  A ,  b  e.  B  |->  C )  =  ( i  e.  A , 
j  e.  B  |->  D ) )
1817oveqd 5867 . . . 4  |-  ( ph  ->  ( i ( a  e.  A ,  b  e.  B  |->  C ) j )  =  ( i ( i  e.  A ,  j  e.  B  |->  D ) j ) )
1918eqeq2d 2182 . . 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 2494 . 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 526 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
i  e.  A )
22 simprr 527 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
j  e.  B )
23 fnmpoovd.d . . . . . 6  |-  ( (
ph  /\  i  e.  A  /\  j  e.  B
)  ->  D  e.  U )
24233expb 1199 . . . . 5  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  ->  D  e.  U )
25 eqid 2170 . . . . . 6  |-  ( i  e.  A ,  j  e.  B  |->  D )  =  ( i  e.  A ,  j  e.  B  |->  D )
2625ovmpt4g 5972 . . . . 5  |-  ( ( i  e.  A  /\  j  e.  B  /\  D  e.  U )  ->  ( i ( i  e.  A ,  j  e.  B  |->  D ) j )  =  D )
2721, 22, 24, 26syl3anc 1233 . . . 4  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  B ) )  -> 
( i ( i  e.  A ,  j  e.  B  |->  D ) j )  =  D )
2827eqeq2d 2182 . . 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 2492 . 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 213 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 set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   A.wral 2448    X. cxp 4607    Fn wfn 5191  (class class class)co 5850    e. cmpo 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117
This theorem is referenced by: (None)
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