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Theorem fnmptfvd 5806
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
StepHypRef Expression
1 fnmptfvd.m . . 3  |-  ( ph  ->  M  Fn  A )
2 fnmptfvd.c . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  C  e.  V )
32ralrimiva 2799 . . . 4  |-  ( ph  ->  A. a  e.  A  C  e.  V )
4 eqid 2443 . . . . 5  |-  ( a  e.  A  |->  C )  =  ( a  e.  A  |->  C )
54fnmpt 5537 . . . 4  |-  ( A. a  e.  A  C  e.  V  ->  ( a  e.  A  |->  C )  Fn  A )
63, 5syl 16 . . 3  |-  ( ph  ->  ( a  e.  A  |->  C )  Fn  A
)
7 eqfnfv 5797 . . 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
) ) )
81, 6, 7syl2anc 661 . 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 )
109cbvmptv 4383 . . . . . . 7  |-  ( i  e.  A  |->  D )  =  ( a  e.  A  |->  C )
1110eqcomi 2447 . . . . . 6  |-  ( a  e.  A  |->  C )  =  ( i  e.  A  |->  D )
1211a1i 11 . . . . 5  |-  ( ph  ->  ( a  e.  A  |->  C )  =  ( i  e.  A  |->  D ) )
1312fveq1d 5693 . . . 4  |-  ( ph  ->  ( ( a  e.  A  |->  C ) `  i )  =  ( ( i  e.  A  |->  D ) `  i
) )
1413eqeq2d 2454 . . 3  |-  ( ph  ->  ( ( M `  i )  =  ( ( a  e.  A  |->  C ) `  i
)  <->  ( M `  i )  =  ( ( i  e.  A  |->  D ) `  i
) ) )
1514ralbidv 2735 . 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 461 . . . . 5  |-  ( (
ph  /\  i  e.  A )  ->  i  e.  A )
17 fnmptfvd.d . . . . 5  |-  ( (
ph  /\  i  e.  A )  ->  D  e.  U )
18 eqid 2443 . . . . . 6  |-  ( i  e.  A  |->  D )  =  ( i  e.  A  |->  D )
1918fvmpt2 5781 . . . . 5  |-  ( ( i  e.  A  /\  D  e.  U )  ->  ( ( i  e.  A  |->  D ) `  i )  =  D )
2016, 17, 19syl2anc 661 . . . 4  |-  ( (
ph  /\  i  e.  A )  ->  (
( i  e.  A  |->  D ) `  i
)  =  D )
2120eqeq2d 2454 . . 3  |-  ( (
ph  /\  i  e.  A )  ->  (
( M `  i
)  =  ( ( i  e.  A  |->  D ) `  i )  <-> 
( M `  i
)  =  D ) )
2221ralbidva 2731 . 2  |-  ( ph  ->  ( A. i  e.  A  ( M `  i )  =  ( ( i  e.  A  |->  D ) `  i
)  <->  A. i  e.  A  ( M `  i )  =  D ) )
238, 15, 223bitrd 279 1  |-  ( ph  ->  ( M  =  ( a  e.  A  |->  C )  <->  A. i  e.  A  ( M `  i )  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715    e. cmpt 4350    Fn wfn 5413   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  cramerlem1  18493
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