Theorem cicpropdlem 49054

Description: Lemma for cicpropd 49055. (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 49052	. . 3 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
2	1	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
3		cic1st2ndbr 49053	. . . . 5 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))
4	3	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))
5		cicpropd.1	. . . . . . . . 9 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		cicpropd.2	. . . . . . . . 9 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (compf‘𝐶) = (compf‘𝐷))
9	6, 8	isopropd 49046	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Iso‘𝐶) = (Iso‘𝐷))
10	9	oveqd 7370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) = ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)))
11	10	neeq1d 2984	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅ ↔ ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)) ≠ ∅))
12		eqid 2729	. . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
13		eqid 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		cicrcl2 49048	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → 𝐶 ∈ Cat)
15	3, 14	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝐶 ∈ Cat)
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐶 ∈ Cat)
17		ciclcl 17728	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (1st ‘𝑃) ∈ (Base‘𝐶))
18	14, 17	mpancom 688	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → (1st ‘𝑃) ∈ (Base‘𝐶))
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃) ∈ (Base‘𝐶))
20	19	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃) ∈ (Base‘𝐶))
21		cicrcl 17729	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (2nd ‘𝑃) ∈ (Base‘𝐶))
22	14, 21	mpancom 688	. . . . . . . 8 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) → (2nd ‘𝑃) ∈ (Base‘𝐶))
23	3, 22	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (2nd ‘𝑃) ∈ (Base‘𝐶))
24	23	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (2nd ‘𝑃) ∈ (Base‘𝐶))
25	12, 13, 16, 20, 24	brcic 17724	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅))
26		eqid 2729	. . . . . 6 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27		eqid 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
28	5	homfeqbas 17621	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
30	20, 29	eleqtrd 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃) ∈ (Base‘𝐷))
31	30	elfvexd 6863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ V)
32	6, 8, 16, 31	catpropd 17634	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
33	16, 32	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ Cat)
34	24, 29	eleqtrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (2nd ‘𝑃) ∈ (Base‘𝐷))
35	26, 27, 33, 30, 34	brcic 17724	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃) ↔ ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)) ≠ ∅))
36	11, 25, 35	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ (1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃)))
37	4, 36	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃))
38		df-br 5096	. . 3 ⊢ ((1st ‘𝑃)( ≃𝑐 ‘𝐷)(2nd ‘𝑃) ↔ ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐷))
39	37, 38	sylib 218	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ ∈ ( ≃𝑐 ‘𝐷))
40	2, 39	eqeltrd 2828	1 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3438  ∅c0 4286  ⟨cop 4585   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17139  Catccat 17589  Hom_f chomf 17591  comp_fccomf 17592  Isociso 17672   ≃_𝑐 ccic 17721

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-supp 8101  df-cat 17593  df-cid 17594  df-homf 17595  df-comf 17596  df-sect 17673  df-inv 17674  df-iso 17675  df-cic 17722

This theorem is referenced by:  cicpropd  49055