Theorem cic1st2nd 49705

Description: Reconstruction of a pair of isomorphic objects in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
cic1st2nd	⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)

Proof of Theorem cic1st2nd

Step	Hyp	Ref	Expression
1		elfvdm 6913	. . . 4 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ dom ≃𝑐 )
2		cicfn 49700	. . . . 5 ⊢ ≃𝑐 Fn Cat
3	2	fndmi 6637	. . . 4 ⊢ dom ≃𝑐 = Cat
4	1, 3	eleqtrdi 2879	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
5		relcic 49703	. . 3 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
6	4, 5	syl 18	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → Rel ( ≃𝑐 ‘𝐶))
7		1st2nd 8032	. 2 ⊢ ((Rel ( ≃𝑐 ‘𝐶) ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
8	6, 7	mpancom 700	1 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ⟨cop 4597  dom cdm 5659  Rel wrel 5664  ‘cfv 6534  1^st c1st 7980  2^nd c2nd 7981  Catccat 17716   ≃_𝑐 ccic 17848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-supp 8153  df-inv 17801  df-iso 17802  df-cic 17849

This theorem is referenced by:  cic1st2ndbr  49706  cicpropdlem  49707  oppcciceq  49710