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Theorem exidcl 33805
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 33782 . . . . . . . 8 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
31, 2syl5eq 2697 . . . . . . 7 (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
43eleq2d 2716 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐴𝑋𝐴 ∈ dom dom 𝐺))
53eleq2d 2716 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐵𝑋𝐵 ∈ dom dom 𝐺))
64, 5anbi12d 747 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
76pm5.32i 670 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
8 inss1 3866 . . . . . . 7 (Magma ∩ ExId ) ⊆ Magma
98sseli 3632 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10 eqid 2651 . . . . . . 7 dom dom 𝐺 = dom dom 𝐺
1110clmgmOLD 33780 . . . . . 6 ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
129, 11syl3an1 1399 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13123expb 1285 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
147, 13sylbi 207 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15143impb 1279 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
1633ad2ant1 1102 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → 𝑋 = dom dom 𝐺)
1715, 16eleqtrrd 2733 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  dom cdm 5143  ran crn 5144  (class class class)co 6690   ExId cexid 33773  Magmacmagm 33777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-exid 33774  df-mgmOLD 33778
This theorem is referenced by: (None)
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