Theorem shftfibg 11585

Description: Value of a fiber of the relation F. (Contributed by Jim Kingdon, 15-Aug-2021.)

Assertion

Ref	Expression
shftfibg	|- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )

Proof of Theorem shftfibg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> A e. CC )
2		simp1 1028	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> F e. V )
3		simp3 1030	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> B e. CC )
4		shftfvalg 11583	. . . . . . 7 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
5	4	breqd 4141	. . . . . 6 |- ( ( A e. CC /\ F e. V ) -> ( B ( F shift A ) z <-> B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z ) )
6		vex 2824	. . . . . . 7 |- z e. _V
7		eleq1 2301	. . . . . . . . 9 |- ( x = B -> ( x e. CC <-> B e. CC ) )
8		oveq1 6092	. . . . . . . . . 10 |- ( x = B -> ( x - A ) = ( B - A ) )
9	8	breq1d 4140	. . . . . . . . 9 |- ( x = B -> ( ( x - A ) F y <-> ( B - A ) F y ) )
10	7, 9	anbi12d 477	. . . . . . . 8 |- ( x = B -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( B e. CC /\ ( B - A ) F y ) ) )
11		breq2 4134	. . . . . . . . 9 |- ( y = z -> ( ( B - A ) F y <-> ( B - A ) F z ) )
12	11	anbi2d 468	. . . . . . . 8 |- ( y = z -> ( ( B e. CC /\ ( B - A ) F y ) <-> ( B e. CC /\ ( B - A ) F z ) ) )
13		eqid 2238	. . . . . . . 8 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
14	10, 12, 13	brabg 4411	. . . . . . 7 |- ( ( B e. CC /\ z e. _V ) -> ( B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z <-> ( B e. CC /\ ( B - A ) F z ) ) )
15	6, 14	mpan2 429	. . . . . 6 |- ( B e. CC -> ( B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z <-> ( B e. CC /\ ( B - A ) F z ) ) )
16	5, 15	sylan9bb 466	. . . . 5 |- ( ( ( A e. CC /\ F e. V ) /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B e. CC /\ ( B - A ) F z ) ) )
17	1, 2, 3, 16	syl21anc 1277	. . . 4 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B e. CC /\ ( B - A ) F z ) ) )
18	17	3anibar 1196	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B - A ) F z ) )
19	18	abbidv 2358	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> { z | B ( F shift A ) z } = { z | ( B - A ) F z } )
20		imasng 5152	. . 3 |- ( B e. CC -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
21	20	3ad2ant3 1051	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
22	3, 1	subcld 8637	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
23		imasng 5152	. . 3 |- ( ( B - A ) e. CC -> ( F " { ( B - A ) } ) = { z | ( B - A ) F z } )
24	22, 23	syl 14	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( F " { ( B - A ) } ) = { z | ( B - A ) F z } )
25	19, 21, 24	3eqtr4d 2281	1 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   {csn 3709   class class class wbr 4130   {copab 4191   "cima 4777  (class class class)co 6085   CCcc 8177   - cmin 8497   shift cshi 11579

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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-shft 11580

This theorem is used by: (None)