Theorem clmgmOLD 37858

Description: Obsolete version of mgmcl 18656 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 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem clmgmOLD

Step	Hyp	Ref	Expression
1		clmgmOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
2	1	ismgmOLD 37857	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3		fovcdm 7603	. . . . 5 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
4	3	3exp 1120	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
5	2, 4	biimtrdi 253	. . 3 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))))
6	5	pm2.43i 52	. 2 ⊢ (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
7	6	3imp 1111	1 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   × cxp 5683  dom cdm 5685  ⟶wf 6557  (class class class)co 7431  Magmacmagm 37855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-mgmOLD 37856

This theorem is referenced by:  exidcl  37883