Theorem cic1st2nd 49032

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 6857	. . . 4 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ dom ≃𝑐 )
2		cicfn 49027	. . . . 5 ⊢ ≃𝑐 Fn Cat
3	2	fndmi 6586	. . . 4 ⊢ dom ≃𝑐 = Cat
4	1, 3	eleqtrdi 2838	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
5		relcic 49030	. . 3 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → Rel ( ≃𝑐 ‘𝐶))
7		1st2nd 7974	. 2 ⊢ ((Rel ( ≃𝑐 ‘𝐶) ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
8	6, 7	mpancom 688	1 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4583  dom cdm 5619  Rel wrel 5624  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923  Catccat 17570   ≃_𝑐 ccic 17702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-supp 8094  df-inv 17655  df-iso 17656  df-cic 17703

This theorem is referenced by:  cic1st2ndbr  49033  cicpropdlem  49034  oppcciceq  49037