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Theorem clmgmOLD 33981
 Description: Obsolete version of mgmcl 17466 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem clmgmOLD
StepHypRef Expression
1 clmgmOLD.1 . . . . 5 𝑋 = dom dom 𝐺
21ismgmOLD 33980 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3 fovrn 6970 . . . . 5 ((𝐺:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
433exp 1113 . . . 4 (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
52, 4syl6bi 243 . . 3 (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))))
65pm2.43i 52 . 2 (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
763imp 1102 1 ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   × cxp 5264  dom cdm 5266  ⟶wf 6045  (class class class)co 6814  Magmacmagm 33978 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-mgmOLD 33979 This theorem is referenced by:  exidcl  34006
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