Theorem cictr 17772

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 17769	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
2		cicrcl 17770	. . . . . 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 17770	. . . . 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 610	. . 3 ⊢ (𝐶 ∈ Cat → ((𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → ((𝑅 ∈ (Base‘𝐶) ∧ 𝑆 ∈ (Base‘𝐶)) ∧ 𝑇 ∈ (Base‘𝐶))))
8	7	3impib 1117	. 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 767	. . . . . . . . 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 769	. . . . . . . . 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 17766	. . . . . . 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 17766	. . . . . . 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 633	. . . . . 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 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 17744	. . . . . . . . . . . . . 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 17767	. . . . . . . . . . . . 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 1932	. . . . . . . . . 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 1932	. . . . . . . 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 1117	. 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 1087  ∃wex 1781   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  compcco 17232  Catccat 17630  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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-supp 8111  df-cat 17634  df-cid 17635  df-sect 17714  df-inv 17715  df-iso 17716  df-cic 17763

This theorem is referenced by:  cicer  17773  nzerooringczr  21460  cicerALT  49521