Theorem cictr 17823

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 17820	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
2		cicrcl 17821	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
3	1, 2	jca 511	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)))
4	3	ex 412	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶))))
5		cicrcl 17821	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑇 ∈ (Base‘𝐶))
6	5	ex 412	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑆( ≃𝑐 ‘𝐶)𝑇 → 𝑇 ∈ (Base‘𝐶)))
7	4, 6	anim12d 609	. . 3 ⊢ (𝐶 ∈ Cat → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))))
8	7	3impib 1116	. 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 482	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		simpll 766	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅 ∈ (Base‘𝐶))
14		simplr 768	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑆 ∈ (Base‘𝐶))
16	9, 10, 11, 13, 15	cic 17817	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
17		simprr 772	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑇 ∈ (Base‘𝐶))
18	9, 10, 11, 15, 17	cic 17817	. . . . . . 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 632	. . . . . 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 481	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . 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 17795	. . . . . . . . . . . . . 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 17818	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 1930	. . . . . . . . . 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 1930	. . . . . . . 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 406	. . . . . . 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 240	. . . . 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 412	. . . 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 1116	. 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 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109  ⟨cop 4612   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  compcco 17288  Catccat 17681  Isociso 17764   ≃_𝑐 ccic 17813

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-supp 8165  df-cat 17685  df-cid 17686  df-sect 17765  df-inv 17766  df-iso 17767  df-cic 17814

This theorem is referenced by:  cicer  17824  nzerooringczr  21446  cicerALT  48980