Theorem fnmptfvd 5782

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 2615	. . . 4 |- ( ph -> A. a e. A C e. V )
4		eqid 2232	. . . . 5 |- ( a e. A |-> C ) = ( a e. A |-> C )
5	4	fnmpt 5485	. . . 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 5775	. . 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 411	. 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 4206	. . . . . . 7 |- ( i e. A |-> D ) = ( a e. A |-> C )
11	10	eqcomi 2236	. . . . . 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 5672	. . . 4 |- ( ph -> ( ( a e. A |-> C ) ` i ) = ( ( i e. A |-> D ) ` i ) )
14	13	eqeq2d 2244	. . 3 |- ( ph -> ( ( M ` i ) = ( ( a e. A |-> C ) ` i ) <-> ( M ` i ) = ( ( i e. A |-> D ) ` i ) ) )
15	14	ralbidv 2542	. 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 110	. . . . 5 |- ( ( ph /\ i e. A ) -> i e. A )
17		fnmptfvd.d	. . . . 5 |- ( ( ph /\ i e. A ) -> D e. U )
18		eqid 2232	. . . . . 6 |- ( i e. A |-> D ) = ( i e. A |-> D )
19	18	fvmpt2 5761	. . . . 5 |- ( ( i e. A /\ D e. U ) -> ( ( i e. A |-> D ) ` i ) = D )
20	16, 17, 19	syl2anc 411	. . . 4 |- ( ( ph /\ i e. A ) -> ( ( i e. A |-> D ) ` i ) = D )
21	20	eqeq2d 2244	. . 3 |- ( ( ph /\ i e. A ) -> ( ( M ` i ) = ( ( i e. A |-> D ) ` i ) <-> ( M ` i ) = D ) )
22	21	ralbidva 2538	. 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 214	1 |- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  nninfdcinf  7462  nninfwlporlemd  7463  nninfwlporlem  7464