Theorem cicsym 17433

Description: Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicsym	⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)

Proof of Theorem cicsym

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cicrcl 17432	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
2		ciclcl 17431	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
3		eqid 2738	. . . . 5 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
4		eqid 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		simpl 482	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
6		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
7	6	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑅 ∈ (Base‘𝐶))
8		simpl 482	. . . . . 6 ⊢ ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
9	8	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆 ∈ (Base‘𝐶))
10	3, 4, 5, 7, 9	cic 17428	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
11		eqid 2738	. . . . . . . . . 10 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
12	4, 11, 5, 7, 9, 3	isoval 17394	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = dom (𝑅(Inv‘𝐶)𝑆))
13	4, 11, 5, 9, 7	invsym2 17392	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → ◡(𝑆(Inv‘𝐶)𝑅) = (𝑅(Inv‘𝐶)𝑆))
14	13	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Inv‘𝐶)𝑆) = ◡(𝑆(Inv‘𝐶)𝑅))
15	14	dmeqd 5803	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = dom ◡(𝑆(Inv‘𝐶)𝑅))
16		df-rn 5591	. . . . . . . . . 10 ⊢ ran (𝑆(Inv‘𝐶)𝑅) = dom ◡(𝑆(Inv‘𝐶)𝑅)
17	15, 16	eqtr4di 2797	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
18	12, 17	eqtrd 2778	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
19	18	eleq2d 2824	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ 𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅)))
20		vex 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
21		elrng 5789	. . . . . . . 8 ⊢ (𝑓 ∈ V → (𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
22	20, 21	mp1i 13	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
23	19, 22	bitrd 278	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
24		df-br 5071	. . . . . . . 8 ⊢ (𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
25	24	exbii 1851	. . . . . . 7 ⊢ (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
26		vex 3426	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
27	26, 20	opeldm 5805	. . . . . . . . . 10 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅))
28	4, 11, 5, 9, 7, 3	isoval 17394	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Iso‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅))
29	28	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑆(Inv‘𝐶)𝑅) = (𝑆(Iso‘𝐶)𝑅))
30	29	eleq2d 2824	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) ↔ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)))
31	5	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝐶 ∈ Cat)
32	9	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆 ∈ (Base‘𝐶))
33	7	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑅 ∈ (Base‘𝐶))
34		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅))
35	3, 4, 31, 32, 33, 34	brcici 17429	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆( ≃𝑐 ‘𝐶)𝑅)
36	35	ex 412	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑆(Iso‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
37	30, 36	sylbid 239	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
38	27, 37	syl5com 31	. . . . . . . . 9 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
39	38	exlimiv 1934	. . . . . . . 8 ⊢ (∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
40	39	com12 32	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
41	25, 40	syl5bi 241	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 → 𝑆( ≃𝑐 ‘𝐶)𝑅))
42	23, 41	sylbid 239	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
43	42	exlimdv 1937	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
44	10, 43	sylbid 239	. . 3 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑆( ≃𝑐 ‘𝐶)𝑅))
45	44	impancom 451	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
46	1, 2, 45	mp2and 695	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   class class class wbr 5070  ^◡ccnv 5579  dom cdm 5580  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Catccat 17290  Invcinv 17374  Isociso 17375   ≃_𝑐 ccic 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-supp 7949  df-sect 17376  df-inv 17377  df-iso 17378  df-cic 17425

This theorem is referenced by:  cicer  17435  initoeu2  17647