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Theorem exidresid 37887
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 6044 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
31, 2eqeltrid 2844 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
4 eqid 2736 . . . . . 6 ran 𝐻 = ran 𝐻
54gidval 30532 . . . . 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 1133 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
87adantr 480 . 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 37885 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
1211simprd 495 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
1312adantr 480 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
149, 10, 1exidres 37886 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )
15 elin 3966 . . . . . . 7 (𝐻 ∈ (Magma ∩ ExId ) ↔ (𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ))
16 rngopidOLD 37861 . . . . . . 7 (𝐻 ∈ (Magma ∩ ExId ) → ran 𝐻 = dom dom 𝐻)
1715, 16sylbir 235 . . . . . 6 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ran 𝐻 = dom dom 𝐻)
1817ancoms 458 . . . . 5 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
1914, 18sylan 580 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
2013, 19raleqtrrdv 3329 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
2111simpld 494 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
2221adantr 480 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ dom dom 𝐻)
2322, 19eleqtrrd 2843 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ ran 𝐻)
244exidu1 37864 . . . . . . 7 (𝐻 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2515, 24sylbir 235 . . . . . 6 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2625ancoms 458 . . . . 5 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2714, 26sylan 580 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
28 oveq1 7439 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
2928eqeq1d 2738 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
3029ovanraleqv 7456 . . . . 5 (𝑢 = 𝑈 → (∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3130riota2 7414 . . . 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))
3223, 27, 31syl2anc 584 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
3320, 32mpbid 232 . 2 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈)
348, 33eqtrd 2776 1 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  ∃!wreu 3377  Vcvv 3479  cin 3949  wss 3950   × cxp 5682  dom cdm 5684  ran crn 5685  cres 5686  cfv 6560  crio 7388  (class class class)co 7432  GIdcgi 30510   ExId cexid 37852  Magmacmagm 37856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-riota 7389  df-ov 7435  df-gid 30514  df-exid 37853  df-mgmOLD 37857
This theorem is referenced by:  isdrngo2  37966
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