Theorem exidcl 36034

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 36011	. . . . . . . 8 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3	1, 2	eqtrid 2790	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
4	3	eleq2d 2824	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺))
5	3	eleq2d 2824	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺))
6	4, 5	anbi12d 631	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
7	6	pm5.32i 575	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
8		inss1 4162	. . . . . . 7 ⊢ (Magma ∩ ExId ) ⊆ Magma
9	8	sseli 3917	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10		eqid 2738	. . . . . . 7 ⊢ dom dom 𝐺 = dom dom 𝐺
11	10	clmgmOLD 36009	. . . . . 6 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
12	9, 11	syl3an1 1162	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13	12	3expb 1119	. . . 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 1114	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
16	3	3ad2ant1 1132	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑋 = dom dom 𝐺)
17	15, 16	eleqtrrd 2842	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∩ cin 3886  dom cdm 5589  ran crn 5590  (class class class)co 7275   ExId cexid 36002  Magmacmagm 36006

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-exid 36003  df-mgmOLD 36007

This theorem is referenced by: (None)