Theorem shftfibg 11563

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 11561	. . . . . . 7 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
5	4	breqd 4136	. . . . . 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 6082	. . . . . . . . . 10 |- ( x = B -> ( x - A ) = ( B - A ) )
9	8	breq1d 4135	. . . . . . . . 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 4129	. . . . . . . . 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 4406	. . . . . . 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 5147	. . 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 8627	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
23		imasng 5147	. . 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 ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   {csn 3705   class class class wbr 4125   {copab 4186   "cima 4772  (class class class)co 6075   CCcc 8167   - cmin 8487   shift cshi 11557

This theorem was proved from 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-shft 11558

This theorem is referenced by: (None)