Theorem fnmptfvd 5600

Description: A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)

Hypotheses

Ref	Expression
fnmptfvd.m	|- ( ph -> M Fn A )
fnmptfvd.s	|- ( i = a -> D = C )
fnmptfvd.d	|- ( ( ph /\ i e. A ) -> D e. U )
fnmptfvd.c	|- ( ( ph /\ a e. A ) -> C e. V )

Assertion

Ref	Expression
fnmptfvd	|- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = D ) )

Distinct variable groups:   A, a, i   C, i   D, a   M, a, i   U, a, i   V, a, i   ph, a, i

Allowed substitution hints:   C( a)   D( i)

Proof of Theorem fnmptfvd

Step	Hyp	Ref	Expression
1		fnmptfvd.m	. . 3 |- ( ph -> M Fn A )
2		fnmptfvd.c	. . . . 5 |- ( ( ph /\ a e. A ) -> C e. V )
3	2	ralrimiva 2543	. . . 4 |- ( ph -> A. a e. A C e. V )
4		eqid 2170	. . . . 5 |- ( a e. A |-> C ) = ( a e. A |-> C )
5	4	fnmpt 5324	. . . 4 |- ( A. a e. A C e. V -> ( a e. A |-> C ) Fn A )
6	3, 5	syl 14	. . 3 |- ( ph -> ( a e. A |-> C ) Fn A )
7		eqfnfv 5593	. . 3 |- ( ( M Fn A /\ ( a e. A |-> C ) Fn A ) -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = ( ( a e. A |-> C ) ` i ) ) )
8	1, 6, 7	syl2anc 409	. 2 |- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = ( ( a e. A |-> C ) ` i ) ) )
9		fnmptfvd.s	. . . . . . . 8 |- ( i = a -> D = C )
10	9	cbvmptv 4085	. . . . . . 7 |- ( i e. A |-> D ) = ( a e. A |-> C )
11	10	eqcomi 2174	. . . . . 6 |- ( a e. A |-> C ) = ( i e. A |-> D )
12	11	a1i 9	. . . . 5 |- ( ph -> ( a e. A |-> C ) = ( i e. A |-> D ) )
13	12	fveq1d 5498	. . . 4 |- ( ph -> ( ( a e. A |-> C ) ` i ) = ( ( i e. A |-> D ) ` i ) )
14	13	eqeq2d 2182	. . 3 |- ( ph -> ( ( M ` i ) = ( ( a e. A |-> C ) ` i ) <-> ( M ` i ) = ( ( i e. A |-> D ) ` i ) ) )
15	14	ralbidv 2470	. 2 |- ( ph -> ( A. i e. A ( M ` i ) = ( ( a e. A |-> C ) ` i ) <-> A. i e. A ( M ` i ) = ( ( i e. A |-> D ) ` i ) ) )
16		simpr 109	. . . . 5 |- ( ( ph /\ i e. A ) -> i e. A )
17		fnmptfvd.d	. . . . 5 |- ( ( ph /\ i e. A ) -> D e. U )
18		eqid 2170	. . . . . 6 |- ( i e. A |-> D ) = ( i e. A |-> D )
19	18	fvmpt2 5579	. . . . 5 |- ( ( i e. A /\ D e. U ) -> ( ( i e. A |-> D ) ` i ) = D )
20	16, 17, 19	syl2anc 409	. . . 4 |- ( ( ph /\ i e. A ) -> ( ( i e. A |-> D ) ` i ) = D )
21	20	eqeq2d 2182	. . 3 |- ( ( ph /\ i e. A ) -> ( ( M ` i ) = ( ( i e. A |-> D ) ` i ) <-> ( M ` i ) = D ) )
22	21	ralbidva 2466	. 2 |- ( ph -> ( A. i e. A ( M ` i ) = ( ( i e. A |-> D ) ` i ) <-> A. i e. A ( M ` i ) = D ) )
23	8, 15, 22	3bitrd 213	1 |- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198

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-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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  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-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  nninfdcinf  7147  nninfwlporlemd  7148  nninfwlporlem  7149