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Theorem exidresid 34003
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 5583 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
31, 2syl5eqel 2853 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
4 eqid 2770 . . . . . 6 ran 𝐻 = ran 𝐻
54gidval 27700 . . . . 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 1126 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (GId‘𝐻) = (𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
87adantr 466 . 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 34001 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
1211simprd 477 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
1312adantr 466 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
149, 10, 1exidres 34002 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )
15 elin 3945 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) ↔ (𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ))
16 rngopidOLD 33977 . . . . . . . 8 (𝐻 ∈ (Magma ∩ ExId ) → ran 𝐻 = dom dom 𝐻)
1715, 16sylbir 225 . . . . . . 7 ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ran 𝐻 = dom dom 𝐻)
1817ancoms 455 . . . . . 6 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
1914, 18sylan 561 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
2019raleqdv 3292 . . . 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 476 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
2322adantr 466 . . . . 5 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ dom dom 𝐻)
2423, 19eleqtrrd 2852 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ ran 𝐻)
254exidu1 33980 . . . . . . 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 455 . . . . 5 ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
2814, 27sylan 561 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
29 oveq1 6799 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
3029eqeq1d 2772 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
31 oveq2 6800 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥𝐻𝑢) = (𝑥𝐻𝑈))
3231eqeq1d 2772 . . . . . . 7 (𝑢 = 𝑈 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝑥𝐻𝑈) = 𝑥))
3330, 32anbi12d 608 . . . . . 6 (𝑢 = 𝑈 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3433ralbidv 3134 . . . . 5 (𝑢 = 𝑈 → (∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
3534riota2 6775 . . . 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 565 . . 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 2804 1 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  ∃!wreu 3062  Vcvv 3349  cin 3720  wss 3721   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  cfv 6031  crio 6752  (class class class)co 6792  GIdcgi 27678   ExId cexid 33968  Magmacmagm 33972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-riota 6753  df-ov 6795  df-gid 27682  df-exid 33969  df-mgmOLD 33973
This theorem is referenced by:  isdrngo2  34082
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