Theorem cic1st2nd 49079

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 6851	. . . 4 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ dom ≃𝑐 )
2		cicfn 49074	. . . . 5 ⊢ ≃𝑐 Fn Cat
3	2	fndmi 6580	. . . 4 ⊢ dom ≃𝑐 = Cat
4	1, 3	eleqtrdi 2841	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
5		relcic 49077	. . 3 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → Rel ( ≃𝑐 ‘𝐶))
7		1st2nd 7966	. 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 1541   ∈ wcel 2111  ⟨cop 4577  dom cdm 5611  Rel wrel 5616  ‘cfv 6476  1^st c1st 7914  2^nd c2nd 7915  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-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-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-supp 8086  df-inv 17650  df-iso 17651  df-cic 17698

This theorem is referenced by:  cic1st2ndbr  49080  cicpropdlem  49081  oppcciceq  49084