Theorem grpoidval 28292

Description: Lemma for grpoidcl 28293 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 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (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.

Step	Hyp	Ref	Expression
1		grpoidval.2	. 2 ⊢ 𝑈 = (GId‘𝐺)
2		grpoidval.1	. . . 4 ⊢ 𝑋 = ran 𝐺
3	2	gidval 28291	. . 3 ⊢ (𝐺 ∈ GrpOp → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4		simpl 485	. . . . . . . . 9 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
5	4	ralimi 3162	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
6	5	rgenw 3152	. . . . . . 7 ⊢ ∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
7	6	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → ∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
8	2	grpoidinv 28287	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
9		simpl 485	. . . . . . . . 9 ⊢ ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
10	9	ralimi 3162	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
11	10	reximi 3245	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
12	8, 11	syl 17	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
13	2	grpoideu 28288	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
14	7, 12, 13	3jca 1124	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
15		reupick2 4291	. . . . 5 ⊢ (((∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ 𝑢 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
16	14, 15	sylan 582	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
17	16	riotabidva 7135	. . 3 ⊢ (𝐺 ∈ GrpOp → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
18	3, 17	eqtr4d 2861	. 2 ⊢ (𝐺 ∈ GrpOp → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
19	1, 18	syl5eq 2870	1 ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ∃!wreu 3142  ran crn 5558  ‘cfv 6357  ℩crio 7115  (class class class)co 7158  GrpOpcgr 28268  GIdcgi 28269

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 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

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 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-riota 7116  df-ov 7161  df-grpo 28272  df-gid 28273

This theorem is referenced by:  grpoidcl  28293  grpoidinv2  28294  cnidOLD  28361  hilid  28940