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Theorem exidresid 33310
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 5401 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
31, 2syl5eqel 2702 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
4 eqid 2621 . . . . . 6 ran 𝐻 = ran 𝐻
54gidval 27215 . . . . 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 1080 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
87adantr 481 . 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 33308 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
1211simprd 479 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
1312adantr 481 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
149, 10, 1exidres 33309 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )
15 elin 3774 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) ↔ (𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ))
16 rngopidOLD 33284 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) → ran 𝐻 = dom dom 𝐻)
1715, 16sylbir 225 . . . . . . 7 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ran 𝐻 = dom dom 𝐻)
1817ancoms 469 . . . . . 6 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
1914, 18sylan 488 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
2019raleqdv 3133 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
2113, 20mpbird 247 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
2211simpld 475 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
2322adantr 481 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ dom dom 𝐻)
2423, 19eleqtrrd 2701 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ ran 𝐻)
254exidu1 33287 . . . . . . 7 (𝐻 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2615, 25sylbir 225 . . . . . 6 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2726ancoms 469 . . . . 5 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2814, 27sylan 488 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
29 oveq1 6611 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
3029eqeq1d 2623 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
31 oveq2 6612 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥𝐻𝑢) = (𝑥𝐻𝑈))
3231eqeq1d 2623 . . . . . . 7 (𝑢 = 𝑈 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝑥𝐻𝑈) = 𝑥))
3330, 32anbi12d 746 . . . . . 6 (𝑢 = 𝑈 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3433ralbidv 2980 . . . . 5 (𝑢 = 𝑈 → (∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3534riota2 6587 . . . 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))
3624, 28, 35syl2anc 692 . . 3 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
3721, 36mpbid 222 . 2 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈)
388, 37eqtrd 2655 1 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ∃!wreu 2909  Vcvv 3186  cin 3554  wss 3555   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076  cfv 5847  crio 6564  (class class class)co 6604  GIdcgi 27193   ExId cexid 33275  Magmacmagm 33279
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 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-riota 6565  df-ov 6607  df-gid 27197  df-exid 33276  df-mgmOLD 33280
This theorem is referenced by:  isdrngo2  33389
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