Theorem cicsym 17773

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 17772	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
2		ciclcl 17771	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
3		eqid 2730	. . . . 5 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
4		eqid 2730	. . . . 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 17768	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
11		eqid 2730	. . . . . . . . . 10 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
12	4, 11, 5, 7, 9, 3	isoval 17734	. . . . . . . . 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 17732	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → ◡(𝑆(Inv‘𝐶)𝑅) = (𝑅(Inv‘𝐶)𝑆))
14	13	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Inv‘𝐶)𝑆) = ◡(𝑆(Inv‘𝐶)𝑅))
15	14	dmeqd 5872	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = dom ◡(𝑆(Inv‘𝐶)𝑅))
16		df-rn 5652	. . . . . . . . . 10 ⊢ ran (𝑆(Inv‘𝐶)𝑅) = dom ◡(𝑆(Inv‘𝐶)𝑅)
17	15, 16	eqtr4di 2783	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
18	12, 17	eqtrd 2765	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
19	18	eleq2d 2815	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ 𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅)))
20		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
21		elrng 5858	. . . . . . . 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 5111	. . . . . . . 8 ⊢ (𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
25	24	exbii 1848	. . . . . . 7 ⊢ (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ∃𝑔⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
26		vex 3454	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
27	26, 20	opeldm 5874	. . . . . . . . . 10 ⊢ (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅))
28	4, 11, 5, 9, 7, 3	isoval 17734	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Iso‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅))
29	28	eqcomd 2736	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑆(Inv‘𝐶)𝑅) = (𝑆(Iso‘𝐶)𝑅))
30	29	eleq2d 2815	. . . . . . . . . . 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 17769	. . . . . . . . . . . 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 1930	. . . . . . . 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 1933	. . . 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 699	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   class class class wbr 5110  ^◡ccnv 5640  dom cdm 5641  ran crn 5642  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Catccat 17632  Invcinv 17714  Isociso 17715   ≃_𝑐 ccic 17764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-supp 8143  df-sect 17716  df-inv 17717  df-iso 17718  df-cic 17765

This theorem is referenced by:  cicer  17775  initoeu2  17985  oppccic  49037  cicerALT  49039