Theorem ofsubeq0 9300

Description: Function analogue of subeq0 8559. (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 variables x u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . . . 7 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
2	1	ffnd 5537	. . . . . 6 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
3		simp3 1030	. . . . . . 7 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
4	3	ffnd 5537	. . . . . 6 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
5		simp1 1028	. . . . . 6 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
6		inidm 3440	. . . . . 6 |- ( A i^i A ) = A
7		eqidd 2239	. . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
8		eqidd 2239	. . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
9	1	ffvelcdmda 5846	. . . . . . 7 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
10	3	ffvelcdmda 5846	. . . . . . 7 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
11	9, 10	subcld 8644	. . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) - ( G ` x ) ) e. CC )
12	2, 4, 5, 5, 6, 7, 8, 11	ofvalg 6316	. . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
13		c0ex 8326	. . . . . . 7 |- 0 e. _V
14	13	fvconst2 5934	. . . . . 6 |- ( x e. A -> ( ( A X. { 0 } ) ` x ) = 0 )
15	14	adantl 277	. . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( A X. { 0 } ) ` x ) = 0 )
16	12, 15	eqeq12d 2253	. . . 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 ) )
17	9, 10	subeq0ad 8654	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( F ` x ) - ( G ` x ) ) = 0 <-> ( F ` x ) = ( G ` x ) ) )
18	16, 17	bitrd 188	. . 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 ) ) )
19	18	ralbidva 2546	. 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 ) ) )
20		subcl 8532	. . . . . 6 |- ( ( u e. CC /\ v e. CC ) -> ( u - v ) e. CC )
21	20	adantl 277	. . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( u - v ) e. CC )
22	21, 1, 3, 5, 5, 6	off 6319	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
23	22	ffnd 5537	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) Fn A )
24	13	fconst 5591	. . . 4 |- ( A X. { 0 } ) : A --> { 0 }
25		ffn 5536	. . . 4 |- ( ( A X. { 0 } ) : A --> { 0 } -> ( A X. { 0 } ) Fn A )
26	24, 25	ax-mp 5	. . 3 |- ( A X. { 0 } ) Fn A
27		eqfnfv 5809	. . 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 ) ) )
28	23, 26, 27	sylancl 417	. 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 ) ) )
29		eqfnfv 5809	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
30	2, 4, 29	syl2anc 415	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
31	19, 28, 30	3bitr4d 220	1 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( F oF - G ) = ( A X. { 0 } ) <-> F = G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3709   X. cxp 4774   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   oFcof 6303   CCcc 8183   0cc0 8185   - cmin 8504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-sub 8506

This theorem is used by:  psrridmfi  15200