Theorem fnmpt2ovd 7252

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

Step	Hyp	Ref	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 )
3	2	3expb 1266	. . . . 5 |- ( ( ph /\ ( a e. A /\ b e. B ) ) -> C e. V )
4	3	ralrimivva 2971	. . . 4 |- ( ph -> A. a e. A A. b e. B C e. V )
5		eqid 2622	. . . . 5 |- ( a e. A , b e. B |-> C ) = ( a e. A , b e. B |-> C )
6	5	fnmpt2 7238	. . . 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 17	. . 3 |- ( ph -> ( a e. A , b e. B |-> C ) Fn ( A X. B ) )
8		eqfnov2 6767	. . 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 693	. 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 2764	. . . . . . . 8 |- F/_ a D
11		nfcv 2764	. . . . . . . 8 |- F/_ b D
12		nfcv 2764	. . . . . . . 8 |- F/_ i C
13		nfcv 2764	. . . . . . . 8 |- F/_ j C
14		fnmpt2ovd.s	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> D = C )
15	10, 11, 12, 13, 14	cbvmpt2 6734	. . . . . . 7 |- ( i e. A , j e. B |-> D ) = ( a e. A , b e. B |-> C )
16	15	eqcomi 2631	. . . . . 6 |- ( a e. A , b e. B |-> C ) = ( i e. A , j e. B |-> D )
17	16	a1i 11	. . . . 5 |- ( ph -> ( a e. A , b e. B |-> C ) = ( i e. A , j e. B |-> D ) )
18	17	oveqd 6667	. . . 4 |- ( ph -> ( i ( a e. A , b e. B |-> C ) j ) = ( i ( i e. A , j e. B |-> D ) j ) )
19	18	eqeq2d 2632	. . 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 2989	. 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 794	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> i e. A )
22		simprr 796	. . . . 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 )
24	23	3expb 1266	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> D e. U )
25		eqid 2622	. . . . . 6 |- ( i e. A , j e. B |-> D ) = ( i e. A , j e. B |-> D )
26	25	ovmpt4g 6783	. . . . 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 1326	. . . 4 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> ( i ( i e. A , j e. B |-> D ) j ) = D )
28	27	eqeq2d 2632	. . 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 2988	. 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 294	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   X. cxp 5112   Fn wfn 5883  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mpt2frlmd  20116