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Theorem grpoidval 28275
Description: Lemma for grpoidcl 28276 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1 𝑋 = ran 𝐺
grpoidval.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpoidval (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
Distinct variable groups:   𝑥,𝑢,𝐺   𝑢,𝑈,𝑥   𝑢,𝑋,𝑥

Proof of Theorem grpoidval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpoidval.2 . 2 𝑈 = (GId‘𝐺)
2 grpoidval.1 . . . 4 𝑋 = ran 𝐺
32gidval 28274 . . 3 (𝐺 ∈ GrpOp → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4 simpl 485 . . . . . . . . 9 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
54ralimi 3147 . . . . . . . 8 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
65rgenw 3137 . . . . . . 7 𝑢𝑋 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
76a1i 11 . . . . . 6 (𝐺 ∈ GrpOp → ∀𝑢𝑋 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
82grpoidinv 28270 . . . . . . 7 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
9 simpl 485 . . . . . . . . 9 ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
109ralimi 3147 . . . . . . . 8 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
1110reximi 3230 . . . . . . 7 (∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
128, 11syl 17 . . . . . 6 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
132grpoideu 28271 . . . . . 6 (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
147, 12, 133jca 1124 . . . . 5 (𝐺 ∈ GrpOp → (∀𝑢𝑋 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
15 reupick2 4267 . . . . 5 (((∀𝑢𝑋 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ 𝑢𝑋) → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1614, 15sylan 582 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1716riotabidva 7110 . . 3 (𝐺 ∈ GrpOp → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
183, 17eqtr4d 2858 . 2 (𝐺 ∈ GrpOp → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
191, 18syl5eq 2867 1 (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  wrex 3126  ∃!wreu 3127  ran crn 5532  cfv 6331  crio 7090  (class class class)co 7133  GrpOpcgr 28251  GIdcgi 28252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fo 6337  df-fv 6339  df-riota 7091  df-ov 7136  df-grpo 28255  df-gid 28256
This theorem is referenced by:  grpoidcl  28276  grpoidinv2  28277  cnidOLD  28344  hilid  28923
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