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Theorem iorlid 33328
Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
iorlid.1 𝑋 = ran 𝐺
iorlid.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
iorlid (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)

Proof of Theorem iorlid
Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iorlid.1 . . 3 𝑋 = ran 𝐺
2 iorlid.2 . . 3 𝑈 = (GId‘𝐺)
31, 2idrval 33327 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
41exidu1 33326 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
5 riotacl 6590 . . 3 (∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋)
64, 5syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋)
73, 6eqeltrd 2698 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  ∃!wreu 2910  cin 3559  ran crn 5085  cfv 5857  crio 6575  (class class class)co 6615  GIdcgi 27232   ExId cexid 33314  Magmacmagm 33318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-riota 6576  df-ov 6618  df-gid 27236  df-exid 33315  df-mgmOLD 33319
This theorem is referenced by:  cmpidelt  33329  rngo1cl  33409
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