Theorem cicsym 16287

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 16286	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
2		ciclcl 16285	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
3		eqid 2610	. . . . 5 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
4		eqid 2610	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		simpl 472	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
6		simpr 476	. . . . . 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 472	. . . . . 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 16282	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
11		eqid 2610	. . . . . . . . . . . 12 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
12	4, 11, 5, 7, 9, 3	isoval 16248	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = dom (𝑅(Inv‘𝐶)𝑆))
13	4, 11, 5, 9, 7	invsym2 16246	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → ◡(𝑆(Inv‘𝐶)𝑅) = (𝑅(Inv‘𝐶)𝑆))
14	13	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Inv‘𝐶)𝑆) = ◡(𝑆(Inv‘𝐶)𝑅))
15	14	dmeqd 5248	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = dom ◡(𝑆(Inv‘𝐶)𝑅))
16		df-rn 5049	. . . . . . . . . . . 12 ⊢ ran (𝑆(Inv‘𝐶)𝑅) = dom ◡(𝑆(Inv‘𝐶)𝑅)
17	15, 16	syl6eqr 2662	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
18	12, 17	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
19	18	eleq2d 2673	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ 𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅)))
20		vex 3176	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
21		elrng 5236	. . . . . . . . . 10 ⊢ (𝑓 ∈ V → (𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
22	20, 21	mp1i 13	. . . . . . . . 9 ⊢ ((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 267	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
24		df-br 4584	. . . . . . . . . 10 ⊢ (𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
25	24	exbii 1764	. . . . . . . . 9 ⊢ (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
26		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
27	26, 20	opeldm 5250	. . . . . . . . . . . 12 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅))
28	4, 11, 5, 9, 7, 3	isoval 16248	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Iso‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅))
29	28	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑆(Inv‘𝐶)𝑅) = (𝑆(Iso‘𝐶)𝑅))
30	29	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) ↔ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)))
31	5	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝐶 ∈ Cat)
32	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆 ∈ (Base‘𝐶))
33	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑅 ∈ (Base‘𝐶))
34		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((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 16283	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆( ≃𝑐 ‘𝐶)𝑅)
36	35	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑆(Iso‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
37	30, 36	sylbid 229	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
38	37	com12 32	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
39	27, 38	syl 17	. . . . . . . . . . 11 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
40	39	exlimiv 1845	. . . . . . . . . 10 ⊢ (∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
41	40	com12 32	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
42	25, 41	syl5bi 231	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 → 𝑆( ≃𝑐 ‘𝐶)𝑅))
43	23, 42	sylbid 229	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
44	43	com12 32	. . . . . 6 ⊢ (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
45	44	exlimiv 1845	. . . . 5 ⊢ (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
46	45	com12 32	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
47	10, 46	sylbid 229	. . 3 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑆( ≃𝑐 ‘𝐶)𝑅))
48	47	impancom 455	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
49	1, 2, 48	mp2and 711	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148  Invcinv 16228  Isociso 16229   ≃_𝑐 ccic 16278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-supp 7183  df-sect 16230  df-inv 16231  df-iso 16232  df-cic 16279

This theorem is referenced by:  cicer  16289  initoeu2  16489