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Theorem ofsubeq0 10311
Description: Function analog of subeq0 9627. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubeq0  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  oF  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )

Proof of Theorem ofsubeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 989 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5552 . . . . . . 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 990 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5552 . . . . . . 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 988 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3552 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2438 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2438 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6324 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  oF  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
12 c0ex 9372 . . . . . . 7  |-  0  e.  _V
1312fvconst2 5926 . . . . . 6  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1413adantl 466 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
1511, 14eqeq12d 2451 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  oF  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  =  0 ) )
161ffvelrnda 5836 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
174ffvelrnda 5836 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1816, 17subeq0ad 9721 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F `  x
)  -  ( G `
 x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
1915, 18bitrd 253 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  oF  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
2019ralbidva 2725 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A. x  e.  A  ( ( F  oF  -  G
) `  x )  =  ( ( A  X.  { 0 } ) `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
213, 6, 7, 7, 8offn 6326 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  -  G )  Fn  A )
2212fconst 5589 . . . 4  |-  ( A  X.  { 0 } ) : A --> { 0 }
23 ffn 5552 . . . 4  |-  ( ( A  X.  { 0 } ) : A --> { 0 }  ->  ( A  X.  { 0 } )  Fn  A
)
2422, 23ax-mp 5 . . 3  |-  ( A  X.  { 0 } )  Fn  A
25 eqfnfv 5790 . . 3  |-  ( ( ( F  oF  -  G )  Fn  A  /\  ( A  X.  { 0 } )  Fn  A )  ->  ( ( F  oF  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  oF  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
2621, 24, 25sylancl 662 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  oF  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  oF  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
27 eqfnfv 5790 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
283, 6, 27syl2anc 661 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2920, 26, 283bitr4d 285 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  oF  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2709   {csn 3870    X. cxp 4830    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086    oFcof 6313   CCcc 9272   0cc0 9274    - cmin 9587
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-po 4633  df-so 4634  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589
This theorem is referenced by:  psrridm  17450  psrridmOLD  17451  dv11cn  21442  coeeulem  21661  plydiveu  21733  facth  21741  quotcan  21744  plyexmo  21748  mpaaeu  29450
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