Theorem cicsym 17771

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 17770	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
2		ciclcl 17769	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
3		eqid 2737	. . . . 5 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
4		eqid 2737	. . . . 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 17766	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
11		eqid 2737	. . . . . . . . . 10 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
12	4, 11, 5, 7, 9, 3	isoval 17732	. . . . . . . . 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 17730	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → ◡(𝑆(Inv‘𝐶)𝑅) = (𝑅(Inv‘𝐶)𝑆))
14	13	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Inv‘𝐶)𝑆) = ◡(𝑆(Inv‘𝐶)𝑅))
15	14	dmeqd 5861	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = dom ◡(𝑆(Inv‘𝐶)𝑅))
16		df-rn 5642	. . . . . . . . . 10 ⊢ ran (𝑆(Inv‘𝐶)𝑅) = dom ◡(𝑆(Inv‘𝐶)𝑅)
17	15, 16	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
18	12, 17	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
19	18	eleq2d 2823	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ 𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅)))
20		vex 3434	. . . . . . . 8 ⊢ 𝑓 ∈ V
21		elrng 5847	. . . . . . . 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 279	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
24		df-br 5087	. . . . . . . 8 ⊢ (𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
25	24	exbii 1850	. . . . . . 7 ⊢ (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
26		vex 3434	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
27	26, 20	opeldm 5863	. . . . . . . . . 10 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅))
28	4, 11, 5, 9, 7, 3	isoval 17732	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Iso‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅))
29	28	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑆(Inv‘𝐶)𝑅) = (𝑆(Iso‘𝐶)𝑅))
30	29	eleq2d 2823	. . . . . . . . . . 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 17767	. . . . . . . . . . . 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 240	. . . . . . . . . 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 1932	. . . . . . . 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	biimtrid 242	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 → 𝑆( ≃𝑐 ‘𝐶)𝑅))
42	23, 41	sylbid 240	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
43	42	exlimdv 1935	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅))
44	10, 43	sylbid 240	. . 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 700	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086  ^◡ccnv 5630  dom cdm 5631  ran crn 5632  ‘cfv 6499  (class class class)co 7367  Basecbs 17179  Catccat 17630  Invcinv 17712  Isociso 17713   ≃_𝑐 ccic 17762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-supp 8111  df-sect 17714  df-inv 17715  df-iso 17716  df-cic 17763

This theorem is referenced by:  cicer  17773  initoeu2  17983  oppccic  49513  cicerALT  49515