Theorem cicpropdlem 49975

Description: Lemma for cicpropd 49976. (Contributed by Zhi Wang, 27-Oct-2025.)

Hypotheses

Ref	Expression
cicpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
cicpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

Assertion

Ref	Expression
cicpropdlem	⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷))

Proof of Theorem cicpropdlem

Step	Hyp	Ref	Expression
1		cic1st2nd 49973	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
2	1	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
3		cic1st2ndbr 49974	. . . . 5 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))
4	3	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))
5		cicpropd.1	. . . . . . . . 9 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		cicpropd.2	. . . . . . . . 9 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (compf‘𝐶) = (compf‘𝐷))
9	6, 8	isopropd 49967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Iso‘𝐶) = (Iso‘𝐷))
10	9	oveqd 7430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) = ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)))
11	10	neeq1d 3014	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅ ↔ ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)) ≠ ∅))
12		eqid 2760	. . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
13		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		cicrcl2 49969	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → 𝐶 ∈ Cat)
15	3, 14	syl 18	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
16	15	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐶 ∈ Cat)
17		ciclcl 17891	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (1st ‘𝑃) ∈ (Base‘𝐶))
18	14, 17	mpancom 701	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → (1st ‘𝑃) ∈ (Base‘𝐶))
19	3, 18	syl 18	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃) ∈ (Base‘𝐶))
20	19	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃) ∈ (Base‘𝐶))
21		cicrcl 17892	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (2nd ‘𝑃) ∈ (Base‘𝐶))
22	14, 21	mpancom 701	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → (2nd ‘𝑃) ∈ (Base‘𝐶))
23	3, 22	syl 18	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (2nd ‘𝑃) ∈ (Base‘𝐶))
24	23	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (2nd ‘𝑃) ∈ (Base‘𝐶))
25	12, 13, 16, 20, 24	brcic 17887	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅))
26		eqid 2760	. . . . . 6 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
28	5	homfeqbas 17784	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
29	28	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
30	20, 29	eleqtrd 2862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃) ∈ (Base‘𝐷))
31	30	elfvexd 6914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ V)
32	6, 8, 16, 31	catpropd 17797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
33	16, 32	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ Cat)
34	24, 29	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (2nd ‘𝑃) ∈ (Base‘𝐷))
35	26, 27, 33, 30, 34	brcic 17887	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃) ↔ ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)) ≠ ∅))
36	11, 25, 35	3bitr4d 314	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ (1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃)))
37	4, 36	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃))
38		df-br 5104	. . 3 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃) ↔ ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐷))
39	37, 38	sylib 221	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐷))
40	2, 39	eqeltrd 2860	1 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4279  ⟨cop 4590   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17301  Catccat 17752  Hom_f chomf 17754  comp_fccomf 17755  Isociso 17835   ≃_𝑐 ccic 17884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-supp 8159  df-cat 17756  df-cid 17757  df-homf 17758  df-comf 17759  df-sect 17836  df-inv 17837  df-iso 17838  df-cic 17885

This theorem is used by:  cicpropd  49976