Theorem rngmgmbs4 33730

Description: The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
rngmgmbs4	|- ( ( G : ( X X. X ) --> X /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) -> ran G = X )

Distinct variable groups:   u, G, x   u, X, x

Proof of Theorem rngmgmbs4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3063	. . . . 5 |- ( E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X E. u e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
2		simpl 473	. . . . . . . . 9 |- ( ( ( u G x ) = x /\ ( x G u ) = x ) -> ( u G x ) = x )
3	2	eqcomd 2628	. . . . . . . 8 |- ( ( ( u G x ) = x /\ ( x G u ) = x ) -> x = ( u G x ) )
4		oveq2 6658	. . . . . . . . . . 11 |- ( y = x -> ( u G y ) = ( u G x ) )
5	4	eqeq2d 2632	. . . . . . . . . 10 |- ( y = x -> ( x = ( u G y ) <-> x = ( u G x ) ) )
6	5	rspcev 3309	. . . . . . . . 9 |- ( ( x e. X /\ x = ( u G x ) ) -> E. y e. X x = ( u G y ) )
7	6	ex 450	. . . . . . . 8 |- ( x e. X -> ( x = ( u G x ) -> E. y e. X x = ( u G y ) ) )
8	3, 7	syl5 34	. . . . . . 7 |- ( x e. X -> ( ( ( u G x ) = x /\ ( x G u ) = x ) -> E. y e. X x = ( u G y ) ) )
9	8	reximdv 3016	. . . . . 6 |- ( x e. X -> ( E. u e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> E. u e. X E. y e. X x = ( u G y ) ) )
10	9	ralimia 2950	. . . . 5 |- ( A. x e. X E. u e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X E. u e. X E. y e. X x = ( u G y ) )
11	1, 10	syl 17	. . . 4 |- ( E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X E. u e. X E. y e. X x = ( u G y ) )
12	11	anim2i 593	. . 3 |- ( ( G : ( X X. X ) --> X /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) -> ( G : ( X X. X ) --> X /\ A. x e. X E. u e. X E. y e. X x = ( u G y ) ) )
13		foov 6808	. . 3 |- ( G : ( X X. X ) -onto-> X <-> ( G : ( X X. X ) --> X /\ A. x e. X E. u e. X E. y e. X x = ( u G y ) ) )
14	12, 13	sylibr 224	. 2 |- ( ( G : ( X X. X ) --> X /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) -> G : ( X X. X ) -onto-> X )
15		forn 6118	. 2 |- ( G : ( X X. X ) -onto-> X -> ran G = X )
16	14, 15	syl 17	1 |- ( ( G : ( X X. X ) --> X /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) -> ran G = X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   ran crn 5115   -->wf 5884   -onto->wfo 5886  (class class class)co 6650

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  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-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-ov 6653

This theorem is referenced by:  rngorn1eq  33733