Theorem clmgmOLD 33650

Description: Obsolete version of mgmcl 17245 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
clmgmOLD.1	|- X = dom dom G

Assertion

Ref	Expression
clmgmOLD	|- ( ( G e. Magma /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Proof of Theorem clmgmOLD

Step	Hyp	Ref	Expression
1		clmgmOLD.1	. . . . 5 |- X = dom dom G
2	1	ismgmOLD 33649	. . . 4 |- ( G e. Magma -> ( G e. Magma <-> G : ( X X. X ) --> X ) )
3		fovrn 6804	. . . . 5 |- ( ( G : ( X X. X ) --> X /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
4	3	3exp 1264	. . . 4 |- ( G : ( X X. X ) --> X -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) )
5	2, 4	syl6bi 243	. . 3 |- ( G e. Magma -> ( G e. Magma -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) ) )
6	5	pm2.43i 52	. 2 |- ( G e. Magma -> ( A e. X -> ( B e. X -> ( A G B ) e. X ) ) )
7	6	3imp 1256	1 |- ( ( G e. Magma /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   X. cxp 5112   dom cdm 5114   -->wf 5884  (class class class)co 6650   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-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-br 4654  df-opab 4713  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-fv 5896  df-ov 6653  df-mgmOLD 33648

This theorem is referenced by:  exidcl  33675