Theorem pconpi1 29953

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 29940	. 2 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )
3		eqid 2450	. . . . 5 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
4		eqid 2450	. . . . 5 |- ( J pi1 ( f ` 1 ) ) = ( J pi1 ( f ` 1 ) )
5		eqid 2450	. . . . 5 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
6		eqid 2450	. . . . 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 1010	. . . . . . 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 29941	. . . . . . 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 19941	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` X ) )
11	9, 10	sylib 200	. . . . 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 763	. . . . 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 6296	. . . . . . 7 |- ( x = y -> ( 1 - x ) = ( 1 - y ) )
14	13	fveq2d 5867	. . . . . 6 |- ( x = y -> ( f ` ( 1 - x ) ) = ( f ` ( 1 - y ) ) )
15	14	cbvmptv 4494	. . . . 5 |- ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( f ` ( 1 - y ) ) )
16	3, 4, 5, 6, 11, 12, 15	pi1xfrgim 22082	. . . 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 773	. . . . . . 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 6304	. . . . . 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 2502	. . . . 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 774	. . . . . . 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 6304	. . . . . 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 2502	. . . . 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 6306	. . . 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 2530	. . 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 16927	. . 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 2882	1 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> P ~=g𝑔 Q )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   <.cop 3973   U.cuni 4197   class class class wbr 4401   |-> cmpt 4460   ran crn 4834   ` cfv 5581  (class class class)co 6288   [cec 7358   0cc0 9536   1c1 9537   - cmin 9857   [,]cicc 11635   Basecbs 15114   GrpIso cgim 16914   ~=g_𝑔 cgic 16915   Topctop 19910  TopOnctopon 19911   Cn ccn 20233   IIcii 21900   ~=ph cphtpc 21993   *pcpco 22024   pi1 cpi1 22027  PConcpcon 29935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-ec 7362  df-qs 7366  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-qus 15402  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-grp 16666  df-mulg 16669  df-ghm 16874  df-gim 16916  df-gic 16917  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-pco 22029  df-om1 22030  df-pi1 22032  df-pcon 29937

This theorem is referenced by:  sconpi1  29955