Theorem cictr 17264

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

Assertion

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

Proof of Theorem cictr

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

Step	Hyp	Ref	Expression
1		ciclcl 17261	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
2		cicrcl 17262	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
3	1, 2	jca 515	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)))
4	3	ex 416	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶))))
5		cicrcl 17262	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑇 ∈ (Base‘𝐶))
6	5	ex 416	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑆( ≃𝑐 ‘𝐶)𝑇 → 𝑇 ∈ (Base‘𝐶)))
7	4, 6	anim12d 612	. . 3 ⊢ (𝐶 ∈ Cat → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))))
8	7	3impib 1118	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))
9		eqid 2736	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
10		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 486	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		simpll 767	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
13	12	adantl 485	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅 ∈ (Base‘𝐶))
14		simplr 769	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
15	14	adantl 485	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑆 ∈ (Base‘𝐶))
16	9, 10, 11, 13, 15	cic 17258	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
17		simprr 773	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑇 ∈ (Base‘𝐶))
18	9, 10, 11, 15, 17	cic 17258	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → (𝑆( ≃𝑐 ‘𝐶)𝑇 ↔ ∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇)))
19	16, 18	anbi12d 634	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) ↔ (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇))))
20	11	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝐶 ∈ Cat)
21	13	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑅 ∈ (Base‘𝐶))
22	17	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑇 ∈ (Base‘𝐶))
23		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
24	15	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑆 ∈ (Base‘𝐶))
25		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆))
26		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇))
27	10, 23, 9, 20, 21, 24, 22, 25, 26	isoco 17236	. . . . . . . . . . . . . 14 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → (𝑔(⟨𝑅, 𝑆⟩(comp‘𝐶)𝑇)𝑓) ∈ (𝑅(Iso‘𝐶)𝑇))
28	9, 10, 20, 21, 22, 27	brcici 17259	. . . . . . . . . . . . 13 ⊢ (((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) ∧ (𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))) → 𝑅( ≃𝑐 ‘𝐶)𝑇)
29	28	ex 416	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) ∧ 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇))
30	29	ex 416	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
31	30	exlimiv 1938	. . . . . . . . . 10 ⊢ (∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
32	31	com12 32	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → (∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
33	32	exlimiv 1938	. . . . . . . 8 ⊢ (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → (∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
34	33	imp 410	. . . . . . 7 ⊢ ((∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇)) → ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅( ≃𝑐 ‘𝐶)𝑇))
35	34	com12 32	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → ((∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆(Iso‘𝐶)𝑇)) → 𝑅( ≃𝑐 ‘𝐶)𝑇))
36	19, 35	sylbid 243	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑅( ≃𝑐 ‘𝐶)𝑇))
37	36	ex 416	. . . 4 ⊢ (𝐶 ∈ Cat → (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
38	37	com23 86	. . 3 ⊢ (𝐶 ∈ Cat → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑅( ≃𝑐 ‘𝐶)𝑇)))
39	38	3impib 1118	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑅( ≃𝑐 ‘𝐶)𝑇))
40	8, 39	mpd 15	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑅( ≃𝑐 ‘𝐶)𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089  ∃wex 1787   ∈ wcel 2112  ⟨cop 4533   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  compcco 16761  Catccat 17121  Isociso 17205   ≃_𝑐 ccic 17254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-supp 7882  df-cat 17125  df-cid 17126  df-sect 17206  df-inv 17207  df-iso 17208  df-cic 17255

This theorem is referenced by:  cicer  17265  nzerooringczr  45246