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Theorem ofsubeq0 10534
Description: Function analog of subeq0 9845. (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 996 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5717 . . . . . . 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 997 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5717 . . . . . . 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 995 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3689 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2442 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2442 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6530 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  oF  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
12 c0ex 9588 . . . . . . 7  |-  0  e.  _V
1312fvconst2 6107 . . . . . 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 2463 . . . 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 6012 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
174ffvelrnda 6012 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1816, 17subeq0ad 9941 . . . 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 2877 . 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 6532 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  -  G )  Fn  A )
2212fconst 5757 . . . 4  |-  ( A  X.  { 0 } ) : A --> { 0 }
23 ffn 5717 . . . 4  |-  ( ( A  X.  { 0 } ) : A --> { 0 }  ->  ( A  X.  { 0 } )  Fn  A
)
2422, 23ax-mp 5 . . 3  |-  ( A  X.  { 0 } )  Fn  A
25 eqfnfv 5962 . . 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 5962 . . 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   {csn 4010    X. cxp 4983    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519   CCcc 9488   0cc0 9490    - cmin 9805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-ltxr 9631  df-sub 9807
This theorem is referenced by:  psrridm  17926  psrridmOLD  17927  dv11cn  22268  coeeulem  22487  plydiveu  22559  facth  22567  quotcan  22570  plyexmo  22574  mpaaeu  31068
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