Theorem cicpropdlem 49536

Description: Lemma for cicpropd 49537. (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 49534	. . 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 49535	. . . . 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 49528	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Iso‘𝐶) = (Iso‘𝐷))
10	9	oveqd 7377	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) = ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)))
11	10	neeq1d 2992	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅ ↔ ((1st ‘𝑃)(Iso‘𝐷)(2nd ‘𝑃)) ≠ ∅))
12		eqid 2737	. . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
13		eqid 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		cicrcl2 49530	. . . . . . . 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 17760	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (1st ‘𝑃) ∈ (Base‘𝐶))
18	14, 17	mpancom 689	. . . . . . . 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 17761	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) → (2nd ‘𝑃) ∈ (Base‘𝐶))
22	14, 21	mpancom 689	. . . . . . . 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 17756	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → ((1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃) ↔ ((1st ‘𝑃)(Iso‘𝐶)(2nd ‘𝑃)) ≠ ∅))
26		eqid 2737	. . . . . 6 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27		eqid 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
28	5	homfeqbas 17653	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
30	20, 29	eleqtrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (1st ‘𝑃) ∈ (Base‘𝐷))
31	30	elfvexd 6870	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝐷 ∈ V)
32	6, 8, 16, 31	catpropd 17666	. . . . . . 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 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → (2nd ‘𝑃) ∈ (Base‘𝐷))
35	26, 27, 33, 30, 34	brcic 17756	. . . . 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 5087	. . 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 2837	1 ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ∅c0 4274  ⟨cop 4574   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Catccat 17621  Hom_f chomf 17623  comp_fccomf 17624  Isociso 17704   ≃_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-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-supp 8104  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-sect 17705  df-inv 17706  df-iso 17707  df-cic 17754

This theorem is referenced by:  cicpropd  49537