Theorem fnmpoovd 6112

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

Step	Hyp	Ref	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 )
3	2	3expb 1182	. . . . 5 |- ( ( ph /\ ( a e. A /\ b e. B ) ) -> C e. V )
4	3	ralrimivva 2514	. . . 4 |- ( ph -> A. a e. A A. b e. B C e. V )
5		eqid 2139	. . . . 5 |- ( a e. A , b e. B |-> C ) = ( a e. A , b e. B |-> C )
6	5	fnmpo 6100	. . . 4 |- ( A. a e. A A. b e. B C e. V -> ( a e. A , b e. B |-> C ) Fn ( A X. B ) )
7	4, 6	syl 14	. . 3 |- ( ph -> ( a e. A , b e. B |-> C ) Fn ( A X. B ) )
8		eqfnov2 5878	. . 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 ) ) )
9	1, 7, 8	syl2anc 408	. 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 2281	. . . . . . . 8 |- F/_ a D
11		nfcv 2281	. . . . . . . 8 |- F/_ b D
12		nfcv 2281	. . . . . . . 8 |- F/_ i C
13		nfcv 2281	. . . . . . . 8 |- F/_ j C
14		fnmpoovd.s	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> D = C )
15	10, 11, 12, 13, 14	cbvmpo 5850	. . . . . . 7 |- ( i e. A , j e. B |-> D ) = ( a e. A , b e. B |-> C )
16	15	eqcomi 2143	. . . . . 6 |- ( a e. A , b e. B |-> C ) = ( i e. A , j e. B |-> D )
17	16	a1i 9	. . . . 5 |- ( ph -> ( a e. A , b e. B |-> C ) = ( i e. A , j e. B |-> D ) )
18	17	oveqd 5791	. . . 4 |- ( ph -> ( i ( a e. A , b e. B |-> C ) j ) = ( i ( i e. A , j e. B |-> D ) j ) )
19	18	eqeq2d 2151	. . 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 ) ) )
20	19	2ralbidv 2459	. 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 520	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> i e. A )
22		simprr 521	. . . . 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 )
24	23	3expb 1182	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> D e. U )
25		eqid 2139	. . . . . 6 |- ( i e. A , j e. B |-> D ) = ( i e. A , j e. B |-> D )
26	25	ovmpt4g 5893	. . . . 5 |- ( ( i e. A /\ j e. B /\ D e. U ) -> ( i ( i e. A , j e. B |-> D ) j ) = D )
27	21, 22, 24, 26	syl3anc 1216	. . . 4 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> ( i ( i e. A , j e. B |-> D ) j ) = D )
28	27	eqeq2d 2151	. . 3 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> ( ( i M j ) = ( i ( i e. A , j e. B |-> D ) j ) <-> ( i M j ) = D ) )
29	28	2ralbidva 2457	. 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 ) )
30	9, 20, 29	3bitrd 213	1 |- ( ph -> ( M = ( a e. A , b e. B |-> C ) <-> A. i e. A A. j e. B ( i M j ) = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   X. cxp 4537   Fn wfn 5118  (class class class)co 5774   e. cmpo 5776

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039

This theorem is referenced by: (None)