Theorem cictr 17863

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 17860	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
2		cicrcl 17861	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
3	1, 2	jca 520	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)))
4	3	ex 417	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → (𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶))))
5		cicrcl 17861	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑇 ∈ (Base‘𝐶))
6	5	ex 417	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑆( ≃𝑐 ‘𝐶)𝑇 → 𝑇 ∈ (Base‘𝐶)))
7	4, 6	anim12d 620	. . 3 ⊢ (𝐶 ∈ Cat → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))))
8	7	3impib 1134	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)))
9		eqid 2763	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
10		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 487	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		simpll 778	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
13	12	adantl 486	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑅 ∈ (Base‘𝐶))
14		simplr 780	. . . . . . . . 9 ⊢ (((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
15	14	adantl 486	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑆 ∈ (Base‘𝐶))
16	9, 10, 11, 13, 15	cic 17857	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
17		simprr 784	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))) → 𝑇 ∈ (Base‘𝐶))
18	9, 10, 11, 15, 17	cic 17857	. . . . . . 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 643	. . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
24	15	adantl 486	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 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 17835	. . . . . . . . . . . . . 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 17858	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 1960	. . . . . . . . . 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 33	. . . . . . . . 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 1960	. . . . . . . 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 411	. . . . . . 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 33	. . . . . 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 417	. . . 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 87	. . 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 1134	. 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 16	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑅( ≃𝑐 ‘𝐶)𝑇)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103  ∃wex 1809   ∈ wcel 2143  ⟨cop 4596   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  compcco 17323  Catccat 17721  Isociso 17804   ≃_𝑐 ccic 17853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-supp 8158  df-cat 17725  df-cid 17726  df-sect 17805  df-inv 17806  df-iso 17807  df-cic 17854

This theorem is referenced by:  cicer  17864  nzerooringczr  21611  cicerALT  49801