Theorem pi1bvalqs 16091

Description: Rewrite the value of pi1b in terms of a quotient set.

Hypothesis

Ref	Expression
pi1bvalqs.1	|- X = U.J

Assertion

Ref	Expression
pi1bvalqs	|- ((J e. Top /\ Y e. X) -> (pi1b` <.J, Y>.) = ({f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}/.(~=ph` J)))

Distinct variable groups:   f,J   f,Y

Proof of Theorem pi1bvalqs

Step	Hyp	Ref	Expression
1		fveq1 4680	. . . . . . . . . 10 |- (f = h -> (f` 0) = (h` 0))
2	1	eqeq1d 1892	. . . . . . . . 9 |- (f = h -> ((f` 0) = Y <-> (h` 0) = Y))
3		fveq1 4680	. . . . . . . . . 10 |- (f = h -> (f` 1) = (h` 1))
4	3	eqeq1d 1892	. . . . . . . . 9 |- (f = h -> ((f` 1) = Y <-> (h` 1) = Y))
5	2, 4	anbi12d 690	. . . . . . . 8 |- (f = h -> (((f` 0) = Y /\ (f` 1) = Y) <-> ((h` 0) = Y /\ (h` 1) = Y)))
6	5	elrab 2414	. . . . . . 7 |- (h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)} <-> (h e. (II Cn J) /\ ((h` 0) = Y /\ (h` 1) = Y)))
7	6	anbi1i 539	. . . . . 6 |- ((h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)} /\ g = [h](~=ph` J)) <-> ((h e. (II Cn J) /\ ((h` 0) = Y /\ (h` 1) = Y)) /\ g = [h](~=ph` J)))
8		anass 487	. . . . . 6 |- (((h e. (II Cn J) /\ ((h` 0) = Y /\ (h` 1) = Y)) /\ g = [h](~=ph` J)) <-> (h e. (II Cn J) /\ (((h` 0) = Y /\ (h` 1) = Y) /\ g = [h](~=ph` J))))
9	7, 8	bitr2i 191	. . . . 5 |- ((h e. (II Cn J) /\ (((h` 0) = Y /\ (h` 1) = Y) /\ g = [h](~=ph` J))) <-> (h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)} /\ g = [h](~=ph` J)))
10	9	rexbii2 2132	. . . 4 |- (E.h e. (II Cn J)(((h` 0) = Y /\ (h` 1) = Y) /\ g = [h](~=ph` J)) <-> E.h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}g = [h](~=ph` J))
11	10	a1i 8	. . 3 |- ((J e. Top /\ Y e. X) -> (E.h e. (II Cn J)(((h` 0) = Y /\ (h` 1) = Y) /\ g = [h](~=ph` J)) <-> E.h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}g = [h](~=ph` J)))
12		pi1bvalqs.1	. . . 4 |- X = U.J
13	12	elpi1 16089	. . 3 |- ((J e. Top /\ Y e. X) -> (g e. (pi1b` <.J, Y>.) <-> E.h e. (II Cn J)(((h` 0) = Y /\ (h` 1) = Y) /\ g = [h](~=ph` J))))
14		visset 2295	. . . . 5 |- g e. _V
15	14	elqs 5348	. . . 4 |- (g e. ({f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}/.(~=ph` J)) <-> E.h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}g = [h](~=ph` J))
16	15	a1i 8	. . 3 |- ((J e. Top /\ Y e. X) -> (g e. ({f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}/.(~=ph` J)) <-> E.h e. {f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}g = [h](~=ph` J)))
17	11, 13, 16	3bitr4d 609	. 2 |- ((J e. Top /\ Y e. X) -> (g e. (pi1b` <.J, Y>.) <-> g e. ({f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}/.(~=ph` J))))
18	17	eqrdv 1882	1 |- ((J e. Top /\ Y e. X) -> (pi1b` <.J, Y>.) = ({f e. (II Cn J) | ((f` 0) = Y /\ (f` 1) = Y)}/.(~=ph` J)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  <.cop 3046  U.cuni 3177  ` cfv 3998  (class class class)co 4884  [cec 5316  /.cqs 5317  0cc0 6386  1c1 6387  Topctop 8857   Cn ccn 9028  IIcii 15865  ~=phcphtpc 16044  pi1bcpi1b 16066

This theorem is referenced by:  pi1f 16093

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-ec 5320  df-qs 5323  df-pi1b 16070

Copyright terms: Public domain