Theorem gafo 17729

Description: A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
gaf.1	|- X = ( Base ` G )

Assertion

Ref	Expression
gafo	|- ( .(+) e. ( G GrpAct Y ) -> .(+) : ( X X. Y ) -onto-> Y )

Proof of Theorem gafo

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

Step	Hyp	Ref	Expression
1		gaf.1	. . 3 |- X = ( Base ` G )
2	1	gaf 17728	. 2 |- ( .(+) e. ( G GrpAct Y ) -> .(+) : ( X X. Y ) --> Y )
3		gagrp 17725	. . . . . 6 |- ( .(+) e. ( G GrpAct Y ) -> G e. Grp )
4	3	adantr 481	. . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> G e. Grp )
5		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 17450	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. X )
7	4, 6	syl 17	. . . 4 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> ( 0g ` G ) e. X )
8		simpr 477	. . . 4 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> x e. Y )
9	5	gagrpid 17727	. . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> ( ( 0g ` G ) .(+) x ) = x )
10	9	eqcomd 2628	. . . 4 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> x = ( ( 0g ` G ) .(+) x ) )
11		rspceov 6692	. . . 4 |- ( ( ( 0g ` G ) e. X /\ x e. Y /\ x = ( ( 0g ` G ) .(+) x ) ) -> E. y e. X E. z e. Y x = ( y .(+) z ) )
12	7, 8, 10, 11	syl3anc 1326	. . 3 |- ( ( .(+) e. ( G GrpAct Y ) /\ x e. Y ) -> E. y e. X E. z e. Y x = ( y .(+) z ) )
13	12	ralrimiva 2966	. 2 |- ( .(+) e. ( G GrpAct Y ) -> A. x e. Y E. y e. X E. z e. Y x = ( y .(+) z ) )
14		foov 6808	. 2 |- ( .(+) : ( X X. Y ) -onto-> Y <-> ( .(+) : ( X X. Y ) --> Y /\ A. x e. Y E. y e. X E. z e. Y x = ( y .(+) z ) ) )
15	2, 13, 14	sylanbrc 698	1 |- ( .(+) e. ( G GrpAct Y ) -> .(+) : ( X X. Y ) -onto-> Y )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   GrpAct cga 17722

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ga 17723

This theorem is referenced by: (None)