Theorem fnmpoovd 6324

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 1207	. . . . 5 |- ( ( ph /\ ( a e. A /\ b e. B ) ) -> C e. V )
4	3	ralrimivva 2590	. . . 4 |- ( ph -> A. a e. A A. b e. B C e. V )
5		eqid 2207	. . . . 5 |- ( a e. A , b e. B |-> C ) = ( a e. A , b e. B |-> C )
6	5	fnmpo 6311	. . . 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 6076	. . 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 2350	. . . . . . . 8 |- F/_ a D
11		nfcv 2350	. . . . . . . 8 |- F/_ b D
12		nfcv 2350	. . . . . . . 8 |- F/_ i C
13		nfcv 2350	. . . . . . . 8 |- F/_ j C
14		fnmpoovd.s	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> D = C )
15	10, 11, 12, 13, 14	cbvmpo 6047	. . . . . . 7 |- ( i e. A , j e. B |-> D ) = ( a e. A , b e. B |-> C )
16	15	eqcomi 2211	. . . . . 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 5984	. . . 4 |- ( ph -> ( i ( a e. A , b e. B |-> C ) j ) = ( i ( i e. A , j e. B |-> D ) j ) )
19	18	eqeq2d 2219	. . 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 2532	. 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 1207	. . . . 5 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> D e. U )
25		eqid 2207	. . . . . 6 |- ( i e. A , j e. B |-> D ) = ( i e. A , j e. B |-> D )
26	25	ovmpt4g 6091	. . . . 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 1250	. . . 4 |- ( ( ph /\ ( i e. A /\ j e. B ) ) -> ( i ( i e. A , j e. B |-> D ) j ) = D )
28	27	eqeq2d 2219	. . 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 2530	. 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   X. cxp 4691   Fn wfn 5285  (class class class)co 5967   e. cmpo 5969

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250

This theorem is referenced by: (None)