Theorem cicpropdlem 49011

Description: Lemma for cicpropd 49012. (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 49009	. . 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 49010	. . . . 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 49003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Iso‘𝐶) = (Iso‘𝐷))
10	9	oveqd 7386	. . . . . 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 49005	. . . . . . . 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 17740	. . . . . . . . 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 17741	. . . . . . . . 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 17736	. . . . 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 17633	. . . . . . . . . . 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 6879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ V)
32	6, 8, 16, 31	catpropd 17646	. . . . . . 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 17736	. . . . 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 5103	. . 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 3444  ∅c0 4292  ⟨cop 4591   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604  Isociso 17684   ≃_𝑐 ccic 17733

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-supp 8117  df-cat 17605  df-cid 17606  df-homf 17607  df-comf 17608  df-sect 17685  df-inv 17686  df-iso 17687  df-cic 17734

This theorem is referenced by:  cicpropd  49012