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Theorem iorlid 36367
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 36366 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
41exidu1 36365 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
5 riotacl 7335 . . 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 2834 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  ∃!wreu 3350  cin 3913  ran crn 5638  cfv 6500  crio 7316  (class class class)co 7361  GIdcgi 29481   ExId cexid 36353  Magmacmagm 36357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-gid 29485  df-exid 36354  df-mgmOLD 36358
This theorem is referenced by:  cmpidelt  36368  rngo1cl  36448
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