Theorem shftfibg 9421

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 905	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> A e. CC )
2		simp1 904	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> F e. V )
3		simp3 906	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> B e. CC )
4		shftfvalg 9419	. . . . . . 7 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
5	4	breqd 3775	. . . . . 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 2560	. . . . . . 7 |- z e. _V
7		eleq1 2100	. . . . . . . . 9 |- ( x = B -> ( x e. CC <-> B e. CC ) )
8		oveq1 5519	. . . . . . . . . 10 |- ( x = B -> ( x - A ) = ( B - A ) )
9	8	breq1d 3774	. . . . . . . . 9 |- ( x = B -> ( ( x - A ) F y <-> ( B - A ) F y ) )
10	7, 9	anbi12d 442	. . . . . . . 8 |- ( x = B -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( B e. CC /\ ( B - A ) F y ) ) )
11		breq2 3768	. . . . . . . . 9 |- ( y = z -> ( ( B - A ) F y <-> ( B - A ) F z ) )
12	11	anbi2d 437	. . . . . . . 8 |- ( y = z -> ( ( B e. CC /\ ( B - A ) F y ) <-> ( B e. CC /\ ( B - A ) F z ) ) )
13		eqid 2040	. . . . . . . 8 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
14	10, 12, 13	brabg 4006	. . . . . . 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 401	. . . . . 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 435	. . . . 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 1134	. . . 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 1072	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B - A ) F z ) )
19	18	abbidv 2155	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> { z | B ( F shift A ) z } = { z | ( B - A ) F z } )
20		imasng 4690	. . 3 |- ( B e. CC -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
21	20	3ad2ant3 927	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
22	3, 1	subcld 7322	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
23		imasng 4690	. . 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 2082	1 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   {cab 2026   _Vcvv 2557   {csn 3375   class class class wbr 3764   {copab 3817   "cima 4348  (class class class)co 5512   CCcc 6887   - cmin 7182   shift cshi 9415

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-shft 9416

This theorem is referenced by: (None)