Theorem exidcl 36744

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 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem exidcl

Step	Hyp	Ref	Expression
1		exidcl.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
2		rngopidOLD 36721	. . . . . . . 8 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3	1, 2	eqtrid 2785	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
4	3	eleq2d 2820	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺))
5	3	eleq2d 2820	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺))
6	4, 5	anbi12d 632	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
7	6	pm5.32i 576	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
8		inss1 4229	. . . . . . 7 ⊢ (Magma ∩ ExId ) ⊆ Magma
9	8	sseli 3979	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10		eqid 2733	. . . . . . 7 ⊢ dom dom 𝐺 = dom dom 𝐺
11	10	clmgmOLD 36719	. . . . . 6 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
12	9, 11	syl3an1 1164	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13	12	3expb 1121	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
14	7, 13	sylbi 216	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15	14	3impb 1116	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
16	3	3ad2ant1 1134	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑋 = dom dom 𝐺)
17	15, 16	eleqtrrd 2837	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3948  dom cdm 5677  ran crn 5678  (class class class)co 7409   ExId cexid 36712  Magmacmagm 36716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7412  df-exid 36713  df-mgmOLD 36717

This theorem is referenced by: (None)