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Theorem exidcl 32845
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 32822 . . . . . . . 8 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
31, 2syl5eq 2656 . . . . . . 7 (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺)
43eleq2d 2673 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐴𝑋𝐴 ∈ dom dom 𝐺))
53eleq2d 2673 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐵𝑋𝐵 ∈ dom dom 𝐺))
64, 5anbi12d 743 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
76pm5.32i 667 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)))
8 inss1 3795 . . . . . . 7 (Magma ∩ ExId ) ⊆ Magma
98sseli 3564 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
10 eqid 2610 . . . . . . 7 dom dom 𝐺 = dom dom 𝐺
1110clmgmOLD 32820 . . . . . 6 ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
129, 11syl3an1 1351 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
13123expb 1258 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
147, 13sylbi 206 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
15143impb 1252 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺)
1633ad2ant1 1075 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → 𝑋 = dom dom 𝐺)
1715, 16eleqtrrd 2691 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  cin 3539  dom cdm 5038  ran crn 5039  (class class class)co 6549   ExId cexid 32813  Magmacmagm 32817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-exid 32814  df-mgmOLD 32818
This theorem is referenced by: (None)
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