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Theorem fnmptfvd 5991
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 2881 . . . 4  |-  ( ph  ->  A. a  e.  A  C  e.  V )
4 eqid 2467 . . . . 5  |-  ( a  e.  A  |->  C )  =  ( a  e.  A  |->  C )
54fnmpt 5713 . . . 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 5982 . . 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 4544 . . . . . . 7  |-  ( i  e.  A  |->  D )  =  ( a  e.  A  |->  C )
1110eqcomi 2480 . . . . . 6  |-  ( a  e.  A  |->  C )  =  ( i  e.  A  |->  D )
1211a1i 11 . . . . 5  |-  ( ph  ->  ( a  e.  A  |->  C )  =  ( i  e.  A  |->  D ) )
1312fveq1d 5874 . . . 4  |-  ( ph  ->  ( ( a  e.  A  |->  C ) `  i )  =  ( ( i  e.  A  |->  D ) `  i
) )
1413eqeq2d 2481 . . 3  |-  ( ph  ->  ( ( M `  i )  =  ( ( a  e.  A  |->  C ) `  i
)  <->  ( M `  i )  =  ( ( i  e.  A  |->  D ) `  i
) ) )
1514ralbidv 2906 . 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 2467 . . . . . 6  |-  ( i  e.  A  |->  D )  =  ( i  e.  A  |->  D )
1918fvmpt2 5964 . . . . 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 2481 . . 3  |-  ( (
ph  /\  i  e.  A )  ->  (
( M `  i
)  =  ( ( i  e.  A  |->  D ) `  i )  <-> 
( M `  i
)  =  D ) )
2221ralbidva 2903 . 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 1379    e. wcel 1767   A.wral 2817    |-> cmpt 4511    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  cramerlem1  19056
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