Theorem cic1st2nd 49534

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 6868	. . . 4 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ dom ≃𝑐 )
2		cicfn 49529	. . . . 5 ⊢ ≃𝑐 Fn Cat
3	2	fndmi 6596	. . . 4 ⊢ dom ≃𝑐 = Cat
4	1, 3	eleqtrdi 2847	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
5		relcic 49532	. . 3 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → Rel ( ≃𝑐 ‘𝐶))
7		1st2nd 7985	. 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 4574  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934  Catccat 17621   ≃_𝑐 ccic 17753

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-supp 8104  df-inv 17706  df-iso 17707  df-cic 17754

This theorem is referenced by:  cic1st2ndbr  49535  cicpropdlem  49536  oppcciceq  49539