Theorem clmgmOLD 38502

Description: Obsolete version of mgmcl 18696 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 38501	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3		fovcdm 7580	. . . . 5 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
4	3	3exp 1137	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
5	2, 4	biimtrdi 256	. . 3 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))))
6	5	pm2.43i 53	. 2 ⊢ (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
7	6	3imp 1128	1 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   × cxp 5659  dom cdm 5661  ⟶wf 6532  (class class class)co 7410  Magmacmagm 38499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-mgmOLD 38500

This theorem is referenced by:  exidcl  38527