Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exidres Structured version   Visualization version   GIF version

Theorem exidres 36387
Description: The restriction of a binary operation with identity to a subset containing the identity has an 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
exidres ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )

Proof of Theorem exidres
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.1 . . . 4 𝑋 = ran 𝐺
2 exidres.2 . . . 4 𝑈 = (GId‘𝐺)
3 exidres.3 . . . 4 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
41, 2, 3exidreslem 36386 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
5 oveq1 7368 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
65eqeq1d 2735 . . . . 5 (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
76ovanraleqv 7385 . . . 4 (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
87rspcev 3583 . . 3 ((𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)) → ∃𝑢 ∈ dom dom 𝐻𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
94, 8syl 17 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∃𝑢 ∈ dom dom 𝐻𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
10 resexg 5987 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
113, 10eqeltrid 2838 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
12 eqid 2733 . . . . 5 dom dom 𝐻 = dom dom 𝐻
1312isexid 36356 . . . 4 (𝐻 ∈ V → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
1411, 13syl 17 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
15143ad2ant1 1134 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
169, 15mpbird 257 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  Vcvv 3447  cin 3913  wss 3914   × cxp 5635  dom cdm 5637  ran crn 5638  cres 5639  cfv 6500  (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-res 5649  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:  exidresid  36388
  Copyright terms: Public domain W3C validator