Theorem pconnpi1 31219

Description: All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)

Hypotheses

Ref	Expression
pconnpi1.x	|- X = U. J
pconnpi1.p	|- P = ( J pi1 A )
pconnpi1.q	|- Q = ( J pi1 B )
pconnpi1.s	|- S = ( Base ` P )
pconnpi1.t	|- T = ( Base ` Q )

Assertion

Ref	Expression
pconnpi1	|- ( ( J e. PConn /\ A e. X /\ B e. X ) -> P ~=g𝑔 Q )

Proof of Theorem pconnpi1

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

Step	Hyp	Ref	Expression
1		pconnpi1.x	. . 3 |- X = U. J
2	1	pconncn 31206	. 2 |- ( ( J e. PConn /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )
3		eqid 2622	. . . . 5 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
4		eqid 2622	. . . . 5 |- ( J pi1 ( f ` 1 ) ) = ( J pi1 ( f ` 1 ) )
5		eqid 2622	. . . . 5 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
6		eqid 2622	. . . . 5 |- ran ( h e. U. ( Base ` ( J pi1 ( f ` 0 ) ) ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) ( *p ` J ) ( h ( *p ` J ) f ) ) ] ( ~=ph ` J ) >. ) = ran ( h e. U. ( Base ` ( J pi1 ( f ` 0 ) ) ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) ( *p ` J ) ( h ( *p ` J ) f ) ) ] ( ~=ph ` J ) >. )
7		simpl1 1064	. . . . . . 7 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. PConn )
8		pconntop 31207	. . . . . . 7 |- ( J e. PConn -> J e. Top )
9	7, 8	syl 17	. . . . . 6 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. Top )
10	1	toptopon 20722	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` X ) )
11	9, 10	sylib 208	. . . . 5 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. (TopOn ` X ) )
12		simprl 794	. . . . 5 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> f e. ( II Cn J ) )
13		oveq2 6658	. . . . . . 7 |- ( x = y -> ( 1 - x ) = ( 1 - y ) )
14	13	fveq2d 6195	. . . . . 6 |- ( x = y -> ( f ` ( 1 - x ) ) = ( f ` ( 1 - y ) ) )
15	14	cbvmptv 4750	. . . . 5 |- ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( f ` ( 1 - y ) ) )
16	3, 4, 5, 6, 11, 12, 15	pi1xfrgim 22858	. . . 4 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ran ( h e. U. ( Base ` ( J pi1 ( f ` 0 ) ) ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) ( *p ` J ) ( h ( *p ` J ) f ) ) ] ( ~=ph ` J ) >. ) e. ( ( J pi1 ( f ` 0 ) ) GrpIso ( J pi1 ( f ` 1 ) ) ) )
17		simprrl 804	. . . . . . 7 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 0 ) = A )
18	17	oveq2d 6666	. . . . . 6 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( J pi1 ( f ` 0 ) ) = ( J pi1 A ) )
19		pconnpi1.p	. . . . . 6 |- P = ( J pi1 A )
20	18, 19	syl6eqr 2674	. . . . 5 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( J pi1 ( f ` 0 ) ) = P )
21		simprrr 805	. . . . . . 7 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 1 ) = B )
22	21	oveq2d 6666	. . . . . 6 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( J pi1 ( f ` 1 ) ) = ( J pi1 B ) )
23		pconnpi1.q	. . . . . 6 |- Q = ( J pi1 B )
24	22, 23	syl6eqr 2674	. . . . 5 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( J pi1 ( f ` 1 ) ) = Q )
25	20, 24	oveq12d 6668	. . . 4 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( ( J pi1 ( f ` 0 ) ) GrpIso ( J pi1 ( f ` 1 ) ) ) = ( P GrpIso Q ) )
26	16, 25	eleqtrd 2703	. . 3 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ran ( h e. U. ( Base ` ( J pi1 ( f ` 0 ) ) ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) ( *p ` J ) ( h ( *p ` J ) f ) ) ] ( ~=ph ` J ) >. ) e. ( P GrpIso Q ) )
27		brgici 17712	. . 3 |- ( ran ( h e. U. ( Base ` ( J pi1 ( f ` 0 ) ) ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) ( *p ` J ) ( h ( *p ` J ) f ) ) ] ( ~=ph ` J ) >. ) e. ( P GrpIso Q ) -> P ~=g𝑔 Q )
28	26, 27	syl 17	. 2 |- ( ( ( J e. PConn /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> P ~=g𝑔 Q )
29	2, 28	rexlimddv 3035	1 |- ( ( J e. PConn /\ A e. X /\ B e. X ) -> P ~=g𝑔 Q )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   [cec 7740   0cc0 9936   1c1 9937   - cmin 10266   [,]cicc 12178   Basecbs 15857   GrpIso cgim 17699   ~=g_𝑔 cgic 17700   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   IIcii 22678   ~=ph cphtpc 22768   *pcpco 22800   pi1 cpi1 22803  PConncpconn 31201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-ghm 17658  df-gim 17701  df-gic 17702  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-om1 22806  df-pi1 22808  df-pconn 31203

This theorem is referenced by:  sconnpi1  31221