Theorem shftfib 10831

Description: Value of a fiber of the relation F. (Contributed by Mario Carneiro, 4-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

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

Proof of Theorem shftfib

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

Step	Hyp	Ref	Expression
1		shftfval.1	. . . . . . 7 |- F e. _V
2	1	shftfval 10829	. . . . . 6 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3	2	breqd 4014	. . . . 5 |- ( A e. CC -> ( B ( F shift A ) z <-> B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z ) )
4		vex 2740	. . . . . 6 |- z e. _V
5		eleq1 2240	. . . . . . . 8 |- ( x = B -> ( x e. CC <-> B e. CC ) )
6		oveq1 5881	. . . . . . . . 9 |- ( x = B -> ( x - A ) = ( B - A ) )
7	6	breq1d 4013	. . . . . . . 8 |- ( x = B -> ( ( x - A ) F y <-> ( B - A ) F y ) )
8	5, 7	anbi12d 473	. . . . . . 7 |- ( x = B -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( B e. CC /\ ( B - A ) F y ) ) )
9		breq2 4007	. . . . . . . 8 |- ( y = z -> ( ( B - A ) F y <-> ( B - A ) F z ) )
10	9	anbi2d 464	. . . . . . 7 |- ( y = z -> ( ( B e. CC /\ ( B - A ) F y ) <-> ( B e. CC /\ ( B - A ) F z ) ) )
11		eqid 2177	. . . . . . 7 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
12	8, 10, 11	brabg 4269	. . . . . 6 |- ( ( B e. CC /\ z e. _V ) -> ( B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z <-> ( B e. CC /\ ( B - A ) F z ) ) )
13	4, 12	mpan2 425	. . . . 5 |- ( B e. CC -> ( B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z <-> ( B e. CC /\ ( B - A ) F z ) ) )
14	3, 13	sylan9bb 462	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B e. CC /\ ( B - A ) F z ) ) )
15		ibar 301	. . . . 5 |- ( B e. CC -> ( ( B - A ) F z <-> ( B e. CC /\ ( B - A ) F z ) ) )
16	15	adantl 277	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( B - A ) F z <-> ( B e. CC /\ ( B - A ) F z ) ) )
17	14, 16	bitr4d 191	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B ( F shift A ) z <-> ( B - A ) F z ) )
18	17	abbidv 2295	. 2 |- ( ( A e. CC /\ B e. CC ) -> { z | B ( F shift A ) z } = { z | ( B - A ) F z } )
19		imasng 4993	. . 3 |- ( B e. CC -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
20	19	adantl 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = { z | B ( F shift A ) z } )
21		simpr 110	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
22		simpl 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
23	21, 22	subcld 8267	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
24		imasng 4993	. . 3 |- ( ( B - A ) e. CC -> ( F " { ( B - A ) } ) = { z | ( B - A ) F z } )
25	23, 24	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( F " { ( B - A ) } ) = { z | ( B - A ) F z } )
26	18, 20, 25	3eqtr4d 2220	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2737   {csn 3592   class class class wbr 4003   {copab 4063   "cima 4629  (class class class)co 5874   CCcc 7808   - cmin 8127   shift cshi 10822

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-sub 8129  df-shft 10823

This theorem is referenced by:  shftval  10833