Theorem exidcl 38197

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 38174	. . . . . . . 8 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3	1, 2	eqtrid 2783	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
4	3	eleq2d 2822	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺))
5	3	eleq2d 2822	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺))
6	4, 5	anbi12d 633	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
7	6	pm5.32i 574	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)))
8		inss1 4177	. . . . . . 7 ⊢ (Magma ∩ ExId ) ⊆ Magma
9	8	sseli 3917	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10		eqid 2736	. . . . . . 7 ⊢ dom dom 𝐺 = dom dom 𝐺
11	10	clmgmOLD 38172	. . . . . 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 217	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15	14	3impb 1115	. 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 2839	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3888  dom cdm 5631  ran crn 5632  (class class class)co 7367   ExId cexid 38165  Magmacmagm 38169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-ov 7370  df-exid 38166  df-mgmOLD 38170

This theorem is referenced by: (None)