Theorem pconpi1 29748

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

Hypotheses

Ref	Expression
pconpi1.x	|- X = U. J
pconpi1.p	|- P = ( J pi1 A )
pconpi1.q	|- Q = ( J pi1 B )
pconpi1.s	|- S = ( Base ` P )
pconpi1.t	|- T = ( Base ` Q )

Assertion

Ref	Expression
pconpi1	|- ( ( J e. PCon /\ A e. X /\ B e. X ) -> P ~=g𝑔 Q )

Proof of Theorem pconpi1

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

Step	Hyp	Ref	Expression
1		pconpi1.x	. . 3 |- X = U. J
2	1	pconcn 29735	. 2 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )
3		eqid 2429	. . . . 5 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
4		eqid 2429	. . . . 5 |- ( J pi1 ( f ` 1 ) ) = ( J pi1 ( f ` 1 ) )
5		eqid 2429	. . . . 5 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
6		eqid 2429	. . . . 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 1008	. . . . . . 7 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. PCon )
8		pcontop 29736	. . . . . . 7 |- ( J e. PCon -> J e. Top )
9	7, 8	syl 17	. . . . . 6 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. Top )
10	1	toptopon 19879	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` X ) )
11	9, 10	sylib 199	. . . . 5 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. (TopOn ` X ) )
12		simprl 762	. . . . 5 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> f e. ( II Cn J ) )
13		oveq2 6313	. . . . . . 7 |- ( x = y -> ( 1 - x ) = ( 1 - y ) )
14	13	fveq2d 5885	. . . . . 6 |- ( x = y -> ( f ` ( 1 - x ) ) = ( f ` ( 1 - y ) ) )
15	14	cbvmptv 4518	. . . . 5 |- ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( f ` ( 1 - y ) ) )
16	3, 4, 5, 6, 11, 12, 15	pi1xfrgim 21982	. . . 4 |- ( ( ( J e. PCon /\ 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 772	. . . . . . 7 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 0 ) = A )
18	17	oveq2d 6321	. . . . . 6 |- ( ( ( J e. PCon /\ 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		pconpi1.p	. . . . . 6 |- P = ( J pi1 A )
20	18, 19	syl6eqr 2488	. . . . 5 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( J pi1 ( f ` 0 ) ) = P )
21		simprrr 773	. . . . . . 7 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 1 ) = B )
22	21	oveq2d 6321	. . . . . 6 |- ( ( ( J e. PCon /\ 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		pconpi1.q	. . . . . 6 |- Q = ( J pi1 B )
24	22, 23	syl6eqr 2488	. . . . 5 |- ( ( ( J e. PCon /\ 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 6323	. . . 4 |- ( ( ( J e. PCon /\ 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 2519	. . 3 |- ( ( ( J e. PCon /\ 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 16885	. . 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. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> P ~=g𝑔 Q )
29	2, 28	rexlimddv 2928	1 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> P ~=g𝑔 Q )

Syntax hints:   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   <.cop 4008   U.cuni 4222   class class class wbr 4426   |-> cmpt 4484   ran crn 4855   ` cfv 5601  (class class class)co 6305   [cec 7369   0cc0 9538   1c1 9539   - cmin 9859   [,]cicc 11638   Basecbs 15084   GrpIso cgim 16872   ~=g_𝑔 cgic 16873   Topctop 19848  TopOnctopon 19849   Cn ccn 20171   IIcii 21803   ~=ph cphtpc 21893   *pcpco 21924   pi1 cpi1 21927  PConcpcon 29730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-qus 15366  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-mulg 16627  df-ghm 16832  df-gim 16874  df-gic 16875  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-cn 20174  df-cnp 20175  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805  df-htpy 21894  df-phtpy 21895  df-phtpc 21916  df-pco 21929  df-om1 21930  df-pi1 21932  df-pcon 29732

This theorem is referenced by:  sconpi1  29750