Theorem cic1st2nd 49406

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 6876	. . . 4 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ dom ≃𝑐 )
2		cicfn 49401	. . . . 5 ⊢ ≃𝑐 Fn Cat
3	2	fndmi 6604	. . . 4 ⊢ dom ≃𝑐 = Cat
4	1, 3	eleqtrdi 2847	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
5		relcic 49404	. . 3 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → Rel ( ≃𝑐 ‘𝐶))
7		1st2nd 7993	. 2 ⊢ ((Rel ( ≃𝑐 ‘𝐶) ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
8	6, 7	mpancom 689	1 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4588  dom cdm 5632  Rel wrel 5637  ‘cfv 6500  1^st c1st 7941  2^nd c2nd 7942  Catccat 17599   ≃_𝑐 ccic 17731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-supp 8113  df-inv 17684  df-iso 17685  df-cic 17732

This theorem is referenced by:  cic1st2ndbr  49407  cicpropdlem  49408  oppcciceq  49411