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Theorem exidcl 35807
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 35784 . . . . . . . 8 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
31, 2syl5eq 2792 . . . . . . 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 634 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
76pm5.32i 578 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
8 inss1 4159 . . . . . . 7 (Magma ∩ ExId ) ⊆ Magma
98sseli 3913 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10 eqid 2739 . . . . . . 7 dom dom 𝐺 = dom dom 𝐺
1110clmgmOLD 35782 . . . . . 6 ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
129, 11syl3an1 1165 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13123expb 1122 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
147, 13sylbi 220 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15143impb 1117 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
1633ad2ant1 1135 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → 𝑋 = dom dom 𝐺)
1715, 16eleqtrrd 2843 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2112  cin 3882  dom cdm 5568  ran crn 5569  (class class class)co 7234   ExId cexid 35775  Magmacmagm 35779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-fo 6406  df-fv 6408  df-ov 7237  df-exid 35776  df-mgmOLD 35780
This theorem is referenced by: (None)
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