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Theorem clmgmOLD 35194
Description: Obsolete version of mgmcl 17846 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 35193 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
3 fovrn 7303 . . . . 5 ((𝐺:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
433exp 1116 . . . 4 (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
52, 4syl6bi 256 . . 3 (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))))
65pm2.43i 52 . 2 (𝐺 ∈ Magma → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))
763imp 1108 1 ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115   × cxp 5536  dom cdm 5538  wf 6334  (class class class)co 7140  Magmacmagm 35191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-fv 6346  df-ov 7143  df-mgmOLD 35192
This theorem is referenced by:  exidcl  35219
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