Theorem cic1st2ndbr 49025

Description: Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
cic1st2ndbr	⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))

Proof of Theorem cic1st2ndbr

Step	Hyp	Ref	Expression
1		cic1st2nd 49024	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
2		id 22	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 ∈ ( ≃𝑐 ‘𝐶))
3	1, 2	eqeltrrd 2830	. 2 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐶))
4		df-br 5110	. 2 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐶))
5	3, 4	sylibr 234	1 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))

Syntax hints:   → wi 4   ∈ wcel 2109  ⟨cop 4597   class class class wbr 5109  ‘cfv 6513  1^st c1st 7968  2^nd c2nd 7969   ≃_𝑐 ccic 17763

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-supp 8142  df-inv 17716  df-iso 17717  df-cic 17764

This theorem is referenced by:  cicpropdlem  49026  oppcciceq  49029