Theorem pi1val 22837

Description: The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)

Hypotheses

Ref	Expression
pi1val.g	|- G = ( J pi1 Y )
pi1val.1	|- ( ph -> J e. (TopOn ` X ) )
pi1val.2	|- ( ph -> Y e. X )
pi1val.o	|- O = ( J Om1 Y )

Assertion

Ref	Expression
pi1val	|- ( ph -> G = ( O /.s ( ~=ph ` J ) ) )

Proof of Theorem pi1val

Dummy variables j y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1val.g	. 2 |- G = ( J pi1 Y )
2		df-pi1 22808	. . . 4 |- pi1 = ( j e. Top , y e. U. j |-> ( ( j Om1 y ) /.s ( ~=ph ` j ) ) )
3	2	a1i 11	. . 3 |- ( ph -> pi1 = ( j e. Top , y e. U. j |-> ( ( j Om1 y ) /.s ( ~=ph ` j ) ) ) )
4		simprl 794	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> j = J )
5		simprr 796	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> y = Y )
6	4, 5	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( j Om1 y ) = ( J Om1 Y ) )
7		pi1val.o	. . . . 5 |- O = ( J Om1 Y )
8	6, 7	syl6eqr 2674	. . . 4 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( j Om1 y ) = O )
9	4	fveq2d 6195	. . . 4 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( ~=ph ` j ) = ( ~=ph ` J ) )
10	8, 9	oveq12d 6668	. . 3 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( ( j Om1 y ) /.s ( ~=ph ` j ) ) = ( O /.s ( ~=ph ` J ) ) )
11		unieq 4444	. . . . 5 |- ( j = J -> U. j = U. J )
12	11	adantl 482	. . . 4 |- ( ( ph /\ j = J ) -> U. j = U. J )
13		pi1val.1	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
14		toponuni 20719	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	syl 17	. . . . 5 |- ( ph -> X = U. J )
16	15	adantr 481	. . . 4 |- ( ( ph /\ j = J ) -> X = U. J )
17	12, 16	eqtr4d 2659	. . 3 |- ( ( ph /\ j = J ) -> U. j = X )
18		topontop 20718	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
19	13, 18	syl 17	. . 3 |- ( ph -> J e. Top )
20		pi1val.2	. . 3 |- ( ph -> Y e. X )
21		ovexd 6680	. . 3 |- ( ph -> ( O /.s ( ~=ph ` J ) ) e. _V )
22	3, 10, 17, 19, 20, 21	ovmpt2dx 6787	. 2 |- ( ph -> ( J pi1 Y ) = ( O /.s ( ~=ph ` J ) ) )
23	1, 22	syl5eq 2668	1 |- ( ph -> G = ( O /.s ( ~=ph ` J ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   /._s cqus 16165   Topctop 20698  TopOnctopon 20715   ~=ph cphtpc 22768   Om1 comi 22801   pi1 cpi1 22803

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-topon 20716  df-pi1 22808

This theorem is referenced by:  pi1bas  22838  pi1addf  22847  pi1addval  22848  pi1grplem  22849