Theorem exidcl 33675

Description: Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypothesis

Ref	Expression
exidcl.1	|- X = ran G

Assertion

Ref	Expression
exidcl	|- ( ( G e. ( Magma i^i ExId ) /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Proof of Theorem exidcl

Step	Hyp	Ref	Expression
1		exidcl.1	. . . . . . . 8 |- X = ran G
2		rngopidOLD 33652	. . . . . . . 8 |- ( G e. ( Magma i^i ExId ) -> ran G = dom dom G )
3	1, 2	syl5eq 2668	. . . . . . 7 |- ( G e. ( Magma i^i ExId ) -> X = dom dom G )
4	3	eleq2d 2687	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( A e. X <-> A e. dom dom G ) )
5	3	eleq2d 2687	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( B e. X <-> B e. dom dom G ) )
6	4, 5	anbi12d 747	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> ( ( A e. X /\ B e. X ) <-> ( A e. dom dom G /\ B e. dom dom G ) ) )
7	6	pm5.32i 669	. . . 4 |- ( ( G e. ( Magma i^i ExId ) /\ ( A e. X /\ B e. X ) ) <-> ( G e. ( Magma i^i ExId ) /\ ( A e. dom dom G /\ B e. dom dom G ) ) )
8		inss1 3833	. . . . . . 7 |- ( Magma i^i ExId ) C_ Magma
9	8	sseli 3599	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> G e. Magma )
10		eqid 2622	. . . . . . 7 |- dom dom G = dom dom G
11	10	clmgmOLD 33650	. . . . . 6 |- ( ( G e. Magma /\ A e. dom dom G /\ B e. dom dom G ) -> ( A G B ) e. dom dom G )
12	9, 11	syl3an1 1359	. . . . 5 |- ( ( G e. ( Magma i^i ExId ) /\ A e. dom dom G /\ B e. dom dom G ) -> ( A G B ) e. dom dom G )
13	12	3expb 1266	. . . 4 |- ( ( G e. ( Magma i^i ExId ) /\ ( A e. dom dom G /\ B e. dom dom G ) ) -> ( A G B ) e. dom dom G )
14	7, 13	sylbi 207	. . 3 |- ( ( G e. ( Magma i^i ExId ) /\ ( A e. X /\ B e. X ) ) -> ( A G B ) e. dom dom G )
15	14	3impb 1260	. 2 |- ( ( G e. ( Magma i^i ExId ) /\ A e. X /\ B e. X ) -> ( A G B ) e. dom dom G )
16	3	3ad2ant1 1082	. 2 |- ( ( G e. ( Magma i^i ExId ) /\ A e. X /\ B e. X ) -> X = dom dom G )
17	15, 16	eleqtrrd 2704	1 |- ( ( G e. ( Magma i^i ExId ) /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   dom cdm 5114   ran crn 5115  (class class class)co 6650   ExId cexid 33643   Magmacmagm 33647

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-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-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  df-exid 33644  df-mgmOLD 33648

This theorem is referenced by: (None)