Theorem relcic 49535

Description: The set of isomorphic objects is a relation. Simplifies cicer 17764 (see cicerALT 49536). (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
relcic	⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))

Proof of Theorem relcic

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

Step	Hyp	Ref	Expression
1		relopab 5767	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
2	1	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
3		fveq2 6827	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
4	3	neeq1d 2993	. . . . . . 7 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
5	4	rabxp 5666	. . . . . 6 ⊢ {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
6	5	a1i 11	. . . . 5 ⊢ (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 5722	. . . 4 ⊢ (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 258	. . 3 ⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9		isofn 17733	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fvex 6840	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
11		sqxpexg 7698	. . . . . 6 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
12	10, 11	mp1i 13	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13		0ex 5229	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
15		suppvalfn 8108	. . . . 5 ⊢ (((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 1379	. . . 4 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
17	16	releqd 5722	. . 3 ⊢ (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
18	8, 17	mpbird 258	. 2 ⊢ (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19		cicfval 17755	. . 3 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
20	19	releqd 5722	. 2 ⊢ (𝐶 ∈ Cat → (Rel ( ≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
21	18, 20	mpbird 258	1 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {crab 3391  Vcvv 3431  ∅c0 4261  ⟨cop 4561  {copab 5134   × cxp 5616  Rel wrel 5623   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356   supp csupp 8100  Basecbs 17170  Catccat 17621  Isociso 17704   ≃_𝑐 ccic 17753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-supp 8101  df-inv 17706  df-iso 17707  df-cic 17754

This theorem is referenced by:  cicerALT  49536  cic1st2nd  49537