Theorem phtpyfval 16046

Description: Value of the path homotopy function on a topology.

Assertion

Ref	Expression
phtpyfval	|- (J e. Top -> (PHtpy` J) = {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})})

Distinct variable group:   f,J,g,z,h,s

Proof of Theorem phtpyfval

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . 7 |- (x = J -> (II Cn x) = (II Cn J))
2	1	eleq2d 1964	. . . . . 6 |- (x = J -> (f e. (II Cn x) <-> f e. (II Cn J)))
3	1	eleq2d 1964	. . . . . 6 |- (x = J -> (g e. (II Cn x) <-> g e. (II Cn J)))
4	2, 3	anbi12d 690	. . . . 5 |- (x = J -> ((f e. (II Cn x) /\ g e. (II Cn x)) <-> (f e. (II Cn J) /\ g e. (II Cn J))))
5	4	anbi1d 679	. . . 4 |- (x = J -> (((f e. (II Cn x) /\ g e. (II Cn x)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) <-> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1)))))
6		opreq2 4890	. . . . . 6 |- (x = J -> ((II X.t II) Cn x) = ((II X.t II) Cn J))
7		rabeq 2289	. . . . . 6 |- (((II X.t II) Cn x) = ((II X.t II) Cn J) -> {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})
8	6, 7	syl 12	. . . . 5 |- (x = J -> {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})
9	8	eqeq2d 1895	. . . 4 |- (x = J -> (z = {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))} <-> z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}))
10	5, 9	anbi12d 690	. . 3 |- (x = J -> ((((f e. (II Cn x) /\ g e. (II Cn x)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}) <-> (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})))
11	10	oprabbidv 4922	. 2 |- (x = J -> {<.<.f, g>., z>. | (((f e. (II Cn x) /\ g e. (II Cn x)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})} = {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})})
12		df-phtpy 16045	. 2 |- PHtpy = {<.x, y>. | (x e. Top /\ y = {<.<.f, g>., z>. | (((f e. (II Cn x) /\ g e. (II Cn x)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn x) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})})}
13		oprex 4907	. . 3 |- (II Cn J) e. _V
14		moeq 2431	. . . . 5 |- E*z z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}
15	14	moani 1820	. . . 4 |- E*z(((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})
16	15	a1i 8	. . 3 |- ((f e. (II Cn J) /\ g e. (II Cn J)) -> E*z(((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}))
17		anass 487	. . . 4 |- ((((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}) <-> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})))
18	17	oprabbii 4923	. . 3 |- {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})} = {<.<.f, g>., z>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}))}
19	13, 13, 16, 18	oprabex 4948	. 2 |- {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})} e. _V
20	11, 12, 19	fvopab4 4743	1 |- (J e. Top -> (PHtpy` J) = {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E*wmo 1772  A.wral 2105  {crab 2108  ` cfv 3998  (class class class)co 4884  {copab2 4885  0cc0 6386  1c1 6387  [,]cicc 7527  Topctop 8857   X.t ctx 8930   Cn ccn 9028  IIcii 15865  PHtpycphtpy 16043

This theorem is referenced by:  phtpyval 16047

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-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  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-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-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-phtpy 16045

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