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Theorem exidcl 35035
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 35012 . . . . . . . 8 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
31, 2syl5eq 2865 . . . . . . 7 (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
43eleq2d 2895 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐴𝑋𝐴 ∈ dom dom 𝐺))
53eleq2d 2895 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐵𝑋𝐵 ∈ dom dom 𝐺))
64, 5anbi12d 630 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
76pm5.32i 575 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
8 inss1 4202 . . . . . . 7 (Magma ∩ ExId ) ⊆ Magma
98sseli 3960 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10 eqid 2818 . . . . . . 7 dom dom 𝐺 = dom dom 𝐺
1110clmgmOLD 35010 . . . . . 6 ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
129, 11syl3an1 1155 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13123expb 1112 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
147, 13sylbi 218 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15143impb 1107 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
1633ad2ant1 1125 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → 𝑋 = dom dom 𝐺)
1715, 16eleqtrrd 2913 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932  dom cdm 5548  ran crn 5549  (class class class)co 7145   ExId cexid 35003  Magmacmagm 35007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-exid 35004  df-mgmOLD 35008
This theorem is referenced by: (None)
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