Theorem clmgmOLD 35936

Description: Obsolete version of mgmcl 18244 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 35935	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3		fovrn 7420	. . . . 5 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
4	3	3exp 1117	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
5	2, 4	syl6bi 252	. . 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 1109	1 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   × cxp 5578  dom cdm 5580  ⟶wf 6414  (class class class)co 7255  Magmacmagm 35933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-mgmOLD 35934

This theorem is referenced by:  exidcl  35961