Theorem cicerALT 49078

Description: Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem cicerALT

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

Step	Hyp	Ref	Expression
1		relcic 49077	. 2 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
2		cicsym 17706	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥)
3		cictr 17707	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
4	3	3expb 1120	. 2 ⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
5		cicref 17703	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥)
6		ciclcl 17704	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
7	5, 6	impbida 800	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥))
8	1, 2, 4, 7	iserd 8643	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5086  ‘cfv 6476   Er wer 8614  Basecbs 17115  Catccat 17565   ≃_𝑐 ccic 17697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-supp 8086  df-er 8617  df-cat 17569  df-cid 17570  df-sect 17649  df-inv 17650  df-iso 17651  df-cic 17698

This theorem is referenced by: (None)