Theorem cicer 17742

Description: Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicer	⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Proof of Theorem cicer

Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5778	. . . . . 6 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
2	1	a1i 11	. . . . 5 ⊢ (𝐶 ∈ Cat → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
3		fveq2 6842	. . . . . . . . 9 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
4	3	neeq1d 2992	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
5	4	rabxp 5680	. . . . . . 7 ⊢ {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
6	5	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
7	6	releqd 5736	. . . . 5 ⊢ (𝐶 ∈ Cat → (Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}))
8	2, 7	mpbird 257	. . . 4 ⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9		isofn 17711	. . . . . 6 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fvex 6855	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11		sqxpexg 7710	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13		0ex 5254	. . . . . . 7 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
15		suppvalfn 8120	. . . . . 6 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Base‘𝐶) × (Base‘𝐶)) ∈ V ∧ ∅ ∈ V) → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
16	9, 12, 14, 15	syl3anc 1374	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
17	16	releqd 5736	. . . 4 ⊢ (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
18	8, 17	mpbird 257	. . 3 ⊢ (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19		cicfval 17733	. . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
20	19	releqd 5736	. . 3 ⊢ (𝐶 ∈ Cat → (Rel ( ≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
21	18, 20	mpbird 257	. 2 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
22		cicsym 17740	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥)
23		cictr 17741	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
24	23	3expb 1121	. 2 ⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
25		cicref 17737	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥)
26		ciclcl 17738	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	impbida 801	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥))
28	21, 22, 24, 27	iserd 8672	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401  Vcvv 3442  ∅c0 4287  ⟨cop 4588   class class class wbr 5100  {copab 5162   × cxp 5630  Rel wrel 5637   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   supp csupp 8112   Er wer 8642  Basecbs 17148  Catccat 17599  Isociso 17682   ≃_𝑐 ccic 17731

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-supp 8113  df-er 8645  df-cat 17603  df-cid 17604  df-sect 17683  df-inv 17684  df-iso 17685  df-cic 17732

This theorem is referenced by: (None)