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Theorem clmgmOLD 33982
Description: Obsolete version of mgmcl 17467 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 33981 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3 fovrn 6971 . . . . 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 2140   × cxp 5265  dom cdm 5267  wf 6046  (class class class)co 6815  Magmacmagm 33979
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-mgmOLD 33980
This theorem is referenced by:  exidcl  34007
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