Theorem fnmpoovd 6375

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 1228	. . . . 5 |- ( ( ph /\ ( a e. A /\ b e. B ) ) -> C e. V )
4	3	ralrimivva 2612	. . . 4 |- ( ph -> A. a e. A A. b e. B C e. V )
5		eqid 2229	. . . . 5 |- ( a e. A , b e. B |-> C ) = ( a e. A , b e. B |-> C )
6	5	fnmpo 6362	. . . 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 6124	. . 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 411	. 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 2372	. . . . . . . 8 |- F/_ a D
11		nfcv 2372	. . . . . . . 8 |- F/_ b D
12		nfcv 2372	. . . . . . . 8 |- F/_ i C
13		nfcv 2372	. . . . . . . 8 |- F/_ j C
14		fnmpoovd.s	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> D = C )
15	10, 11, 12, 13, 14	cbvmpo 6095	. . . . . . 7 |- ( i e. A , j e. B |-> D ) = ( a e. A , b e. B |-> C )
16	15	eqcomi 2233	. . . . . 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 6030	. . . 4 |- ( ph -> ( i ( a e. A , b e. B |-> C ) j ) = ( i ( i e. A , j e. B |-> D ) j ) )
19	18	eqeq2d 2241	. . 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 2554	. 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 529	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> i e. A )
22		simprr 531	. . . . 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 1228	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> D e. U )
25		eqid 2229	. . . . . 6 |- ( i e. A , j e. B |-> D ) = ( i e. A , j e. B |-> D )
26	25	ovmpt4g 6139	. . . . 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 1271	. . . 4 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> ( i ( i e. A , j e. B |-> D ) j ) = D )
28	27	eqeq2d 2241	. . 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 2552	. 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 214	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 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   X. cxp 4721   Fn wfn 5319  (class class class)co 6013   e. cmpo 6015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299

This theorem is referenced by: (None)