Theorem isphtpc2 16060

Description: The relation "is path homotopic to".

Assertion

Ref	Expression
isphtpc2	|- ((J e. Top /\ G e. A) -> (F(~=ph` J)G <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))

Proof of Theorem isphtpc2

Step	Hyp	Ref	Expression
1		relopab 4104	. . . . 5 |- Rel {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))}
2		phtpcval 16058	. . . . . . 7 |- (J e. Top -> (~=ph` J) = {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))})
3	2	adantr 425	. . . . . 6 |- ((J e. Top /\ G e. A) -> (~=ph` J) = {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))})
4		releq 4071	. . . . . 6 |- ((~=ph` J) = {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))} -> (Rel (~=ph` J) <-> Rel {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))}))
5	3, 4	syl 12	. . . . 5 |- ((J e. Top /\ G e. A) -> (Rel (~=ph` J) <-> Rel {<.f, g>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ (f(PHtpy` J)g) =/= (/))}))
6	1, 5	mpbiri 211	. . . 4 |- ((J e. Top /\ G e. A) -> Rel (~=ph` J))
7		brrelex 4028	. . . . 5 |- ((Rel (~=ph` J) /\ F(~=ph` J)G) -> F e. _V)
8	7	ex 402	. . . 4 |- (Rel (~=ph` J) -> (F(~=ph` J)G -> F e. _V))
9	6, 8	syl 12	. . 3 |- ((J e. Top /\ G e. A) -> (F(~=ph` J)G -> F e. _V))
10		isphtpc 16059	. . . . . . . 8 |- ((J e. Top /\ F e. _V /\ G e. A) -> (F(~=ph` J)G <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
11	10	biimpd 170	. . . . . . 7 |- ((J e. Top /\ F e. _V /\ G e. A) -> (F(~=ph` J)G -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
12	11	3expa 1067	. . . . . 6 |- (((J e. Top /\ F e. _V) /\ G e. A) -> (F(~=ph` J)G -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
13	12	an1rs 547	. . . . 5 |- (((J e. Top /\ G e. A) /\ F e. _V) -> (F(~=ph` J)G -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
14	13	ex 402	. . . 4 |- ((J e. Top /\ G e. A) -> (F e. _V -> (F(~=ph` J)G -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)))))
15	14	com23 36	. . 3 |- ((J e. Top /\ G e. A) -> (F(~=ph` J)G -> (F e. _V -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)))))
16	9, 15	mpdd 57	. 2 |- ((J e. Top /\ G e. A) -> (F(~=ph` J)G -> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
17		simplll 452	. . 3 |- ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F e. (II Cn J))
18		isphtpc 16059	. . . . . . . 8 |- ((J e. Top /\ F e. (II Cn J) /\ G e. A) -> (F(~=ph` J)G <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))
19	18	biimprd 171	. . . . . . 7 |- ((J e. Top /\ F e. (II Cn J) /\ G e. A) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F(~=ph` J)G))
20	19	3expa 1067	. . . . . 6 |- (((J e. Top /\ F e. (II Cn J)) /\ G e. A) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F(~=ph` J)G))
21	20	an1rs 547	. . . . 5 |- (((J e. Top /\ G e. A) /\ F e. (II Cn J)) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F(~=ph` J)G))
22	21	ex 402	. . . 4 |- ((J e. Top /\ G e. A) -> (F e. (II Cn J) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F(~=ph` J)G)))
23	22	com23 36	. . 3 |- ((J e. Top /\ G e. A) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> (F e. (II Cn J) -> F(~=ph` J)G)))
24	17, 23	mpdi 59	. 2 |- ((J e. Top /\ G e. A) -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/)) -> F(~=ph` J)G))
25	16, 24	impbid 574	1 |- ((J e. Top /\ G e. A) -> (F(~=ph` J)G <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) /\ (F(PHtpy` J)G) =/= (/))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875   class class class wbr 3338  {copab 3395  Rel wrel 3991  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387  Topctop 8857   Cn ccn 9028  IIcii 15865  PHtpycphtpy 16043  ~=phcphtpc 16044

This theorem is referenced by:  phtpcdm 16061  reparpht 16065  pcohtpy 16083  pcorev 16087

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-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-rex 2110  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-fv 4014  df-opr 4886  df-phtpc 16057

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