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Theorem exidcl 37863
Description: Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypothesis
Ref Expression
exidcl.1 𝑋 = ran 𝐺
Assertion
Ref Expression
exidcl ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem exidcl
StepHypRef Expression
1 exidcl.1 . . . . . . . 8 𝑋 = ran 𝐺
2 rngopidOLD 37840 . . . . . . . 8 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
31, 2eqtrid 2787 . . . . . . 7 (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
43eleq2d 2825 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐴𝑋𝐴 ∈ dom dom 𝐺))
53eleq2d 2825 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐵𝑋𝐵 ∈ dom dom 𝐺))
64, 5anbi12d 632 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
76pm5.32i 574 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
8 inss1 4245 . . . . . . 7 (Magma ∩ ExId ) ⊆ Magma
98sseli 3991 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10 eqid 2735 . . . . . . 7 dom dom 𝐺 = dom dom 𝐺
1110clmgmOLD 37838 . . . . . 6 ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
129, 11syl3an1 1162 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13123expb 1119 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
147, 13sylbi 217 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15143impb 1114 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
1633ad2ant1 1132 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → 𝑋 = dom dom 𝐺)
1715, 16eleqtrrd 2842 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  dom cdm 5689  ran crn 5690  (class class class)co 7431   ExId cexid 37831  Magmacmagm 37835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-exid 37832  df-mgmOLD 37836
This theorem is referenced by: (None)
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