Theorem shftfibg 11331

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 1022	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> A e. CC )
2		simp1 1021	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> F e. V )
3		simp3 1023	. . . . 5 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> B e. CC )
4		shftfvalg 11329	. . . . . . 7 |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
5	4	breqd 4094	. . . . . 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 2802	. . . . . . 7 |- z e. _V
7		eleq1 2292	. . . . . . . . 9 |- ( x = B -> ( x e. CC <-> B e. CC ) )
8		oveq1 6008	. . . . . . . . . 10 |- ( x = B -> ( x - A ) = ( B - A ) )
9	8	breq1d 4093	. . . . . . . . 9 |- ( x = B -> ( ( x - A ) F y <-> ( B - A ) F y ) )
10	7, 9	anbi12d 473	. . . . . . . 8 |- ( x = B -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( B e. CC /\ ( B - A ) F y ) ) )
11		breq2 4087	. . . . . . . . 9 |- ( y = z -> ( ( B - A ) F y <-> ( B - A ) F z ) )
12	11	anbi2d 464	. . . . . . . 8 |- ( y = z -> ( ( B e. CC /\ ( B - A ) F y ) <-> ( B e. CC /\ ( B - A ) F z ) ) )
13		eqid 2229	. . . . . . . 8 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
14	10, 12, 13	brabg 4357	. . . . . . 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 425	. . . . . 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 462	. . . . 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 1270	. . . 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 1189	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B - A ) F z ) )
19	18	abbidv 2347	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> { z | B ( F shift A ) z } = { z | ( B - A ) F z } )
20		imasng 5093	. . 3 |- ( B e. CC -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
21	20	3ad2ant3 1044	. 2 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
22	3, 1	subcld 8457	. . 3 |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
23		imasng 5093	. . 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 2272	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 1002   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   {csn 3666   class class class wbr 4083   {copab 4144   "cima 4722  (class class class)co 6001   CCcc 7997   - cmin 8317   shift cshi 11325

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319  df-shft 11326

This theorem is referenced by: (None)