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Theorem ofsubeq0 9929
Description: Function analog of subeq0 9259. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubeq0  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )

Proof of Theorem ofsubeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5531 . . . . . . 7  |-  ( F : A --> CC  ->  F  Fn  A )
31, 2syl 16 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
4 simp3 959 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5531 . . . . . . 7  |-  ( G : A --> CC  ->  G  Fn  A )
64, 5syl 16 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
7 simp1 957 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3493 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2388 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2388 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6253 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  o F  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
12 c0ex 9018 . . . . . . 7  |-  0  e.  _V
1312fvconst2 5886 . . . . . 6  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1413adantl 453 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
1511, 14eqeq12d 2401 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  =  0 ) )
161ffvelrnda 5809 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
174ffvelrnda 5809 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1816, 17subeq0ad 9353 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F `  x
)  -  ( G `
 x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
1915, 18bitrd 245 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
2019ralbidva 2665 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A. x  e.  A  ( ( F  o F  -  G
) `  x )  =  ( ( A  X.  { 0 } ) `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
213, 6, 7, 7, 8offn 6255 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  -  G )  Fn  A )
2212fconst 5569 . . . 4  |-  ( A  X.  { 0 } ) : A --> { 0 }
23 ffn 5531 . . . 4  |-  ( ( A  X.  { 0 } ) : A --> { 0 }  ->  ( A  X.  { 0 } )  Fn  A
)
2422, 23ax-mp 8 . . 3  |-  ( A  X.  { 0 } )  Fn  A
25 eqfnfv 5766 . . 3  |-  ( ( ( F  o F  -  G )  Fn  A  /\  ( A  X.  { 0 } )  Fn  A )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
2621, 24, 25sylancl 644 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
27 eqfnfv 5766 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
283, 6, 27syl2anc 643 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2920, 26, 283bitr4d 277 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   {csn 3757    X. cxp 4816    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   0cc0 8923    - cmin 9223
This theorem is referenced by:  psrridm  16395  dv11cn  19752  coeeulem  20010  plydiveu  20082  facth  20090  quotcan  20093  plyexmo  20097  mpaaeu  27024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225
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