Theorem shftfib 11446

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 11444	. . . . . 6 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3	2	breqd 4104	. . . . 5 |- ( A e. CC -> ( B ( F shift A ) z <-> B { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } z ) )
4		vex 2806	. . . . . 6 |- z e. _V
5		eleq1 2294	. . . . . . . 8 |- ( x = B -> ( x e. CC <-> B e. CC ) )
6		oveq1 6035	. . . . . . . . 9 |- ( x = B -> ( x - A ) = ( B - A ) )
7	6	breq1d 4103	. . . . . . . 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 4097	. . . . . . . 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 2231	. . . . . . 7 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
12	8, 10, 11	brabg 4369	. . . . . 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 2350	. 2 |- ( ( A e. CC /\ B e. CC ) -> { z | B ( F shift A ) z } = { z | ( B - A ) F z } )
19		imasng 5108	. . 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 8532	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B - A ) e. CC )
24		imasng 5108	. . 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 2274	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   _Vcvv 2803   {csn 3673   class class class wbr 4093   {copab 4154   "cima 4734  (class class class)co 6028   CCcc 8073   - cmin 8392   shift cshi 11437

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-resscn 8167  ax-1cn 8168  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sub 8394  df-shft 11438

This theorem is referenced by:  shftval  11448