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Theorem exidresid 36747
Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
exidres.1 𝑋 = ran 𝐺
exidres.2 𝑈 = (GId‘𝐺)
exidres.3 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
exidresid (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)

Proof of Theorem exidresid
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.3 . . . . . 6 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
2 resexg 6028 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
31, 2eqeltrid 2838 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
4 eqid 2733 . . . . . 6 ran 𝐻 = ran 𝐻
54gidval 29765 . . . . 5 (𝐻 ∈ V → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
63, 5syl 17 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
763ad2ant1 1134 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
87adantr 482 . 2 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
9 exidres.1 . . . . . . 7 𝑋 = ran 𝐺
10 exidres.2 . . . . . . 7 𝑈 = (GId‘𝐺)
119, 10, 1exidreslem 36745 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
1211simprd 497 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
1312adantr 482 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
149, 10, 1exidres 36746 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )
15 elin 3965 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) ↔ (𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ))
16 rngopidOLD 36721 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) → ran 𝐻 = dom dom 𝐻)
1715, 16sylbir 234 . . . . . . 7 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ran 𝐻 = dom dom 𝐻)
1817ancoms 460 . . . . . 6 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
1914, 18sylan 581 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
2019raleqdv 3326 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
2113, 20mpbird 257 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
2211simpld 496 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
2322adantr 482 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ dom dom 𝐻)
2423, 19eleqtrrd 2837 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ ran 𝐻)
254exidu1 36724 . . . . . . 7 (𝐻 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2615, 25sylbir 234 . . . . . 6 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2726ancoms 460 . . . . 5 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2814, 27sylan 581 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
29 oveq1 7416 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
3029eqeq1d 2735 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
3130ovanraleqv 7433 . . . . 5 (𝑢 = 𝑈 → (∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3231riota2 7391 . . . 4 ((𝑈 ∈ ran 𝐻 ∧ ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
3324, 28, 32syl2anc 585 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
3421, 33mpbid 231 . 2 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈)
358, 34eqtrd 2773 1 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  ∃!wreu 3375  Vcvv 3475  cin 3948  wss 3949   × cxp 5675  dom cdm 5677  ran crn 5678  cres 5679  cfv 6544  crio 7364  (class class class)co 7409  GIdcgi 29743   ExId cexid 36712  Magmacmagm 36716
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-gid 29747  df-exid 36713  df-mgmOLD 36717
This theorem is referenced by:  isdrngo2  36826
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