invpropdlem - Mathbox for Zhi Wang

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Theorem invpropdlem 49964

Description: Lemma for invpropd 49965. (Contributed by Zhi Wang, 27-Oct-2025.)

**Hypotheses**
Ref	Expression
sectpropd.1	⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
sectpropd.2	⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))

**Assertion**
Ref	Expression
invpropdlem	⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷))

Proof of Theorem invpropdlem Dummy variables 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐶))
2		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2760	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		df-inv 17837	. . . . . . . 8 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥))))
5	4	mptrcl 6996	. . . . . . 7 ⊢ (𝑃 ∈ (Inv‘𝐶) → 𝐶 ∈ Cat)
6	5	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝐶 ∈ Cat)
7		eqid 2760	. . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	2, 3, 6, 7	invffval 17847	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥))))
9		df-mpo 7418	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥))) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)))}
10	8, 9	eqtrdi 2811	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Inv‘𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)))})
11	1, 10	eleqtrd 2862	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)))})
12		eloprab1st2nd 49796	. . 3 ⊢ (𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)))} → 𝑃 = ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩)
13	11, 12	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 = ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩)
14		sectpropd.1	. . . . . . . . 9 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
15	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
16		sectpropd.2	. . . . . . . . 9 ⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (comp_f‘𝐶) = (comp_f‘𝐷))
18	15, 17	sectpropd 49963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Sect‘𝐶) = (Sect‘𝐷))
19	18	oveqd 7430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) = ((1^st ‘(1^st ‘𝑃))(Sect‘𝐷)(2^nd ‘(1^st ‘𝑃))))
20	18	oveqd 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))) = ((2^nd ‘(1^st ‘𝑃))(Sect‘𝐷)(1^st ‘(1^st ‘𝑃))))
21	20	cnveqd 5855	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))) = ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐷)(1^st ‘(1^st ‘𝑃))))
22	19, 21	ineq12d 4167	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐷)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐷)(1^st ‘(1^st ‘𝑃)))))
23		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥 ∈ (Base‘𝐶) ↔ (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
24	23	anbi1d 643	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))))
25		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥(Sect‘𝐶)𝑦) = ((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦))
26		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑦(Sect‘𝐶)𝑥) = (𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))
27	26	cnveqd 5855	. . . . . . . . . . 11 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → ^◡(𝑦(Sect‘𝐶)𝑥) = ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))
28	25, 27	ineq12d 4167	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))
29	28	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)) ↔ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))))
30	24, 29	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥))) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))))
31		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑦 ∈ (Base‘𝐶) ↔ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
32	31	anbi2d 642	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))))
33		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → ((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) = ((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))))
34		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))) = ((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))
35	34	cnveqd 5855	. . . . . . . . . . 11 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))) = ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))
36	33, 35	ineq12d 4167	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))
37	36	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))) ↔ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))))
38	32, 37	anbi12d 644	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → ((((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))))
39		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑧 = (2^nd ‘𝑃) → (𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))) ↔ (2^nd ‘𝑃) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))))
40	39	anbi2d 642	. . . . . . . 8 ⊢ (𝑧 = (2^nd ‘𝑃) → ((((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ 𝑧 = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))))
41	30, 38, 40	eloprabi 8060	. . . . . . 7 ⊢ (𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = ((𝑥(Sect‘𝐶)𝑦) ∩ ^◡(𝑦(Sect‘𝐶)𝑥)))} → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))))
42	11, 41	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃))))))
43	42	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (2^nd ‘𝑃) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐶)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐶)(1^st ‘(1^st ‘𝑃)))))
44		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
45		eqid 2760	. . . . . 6 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
46	42	simplld 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
47	15	homfeqbas 17784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
48	46, 47	eleqtrd 2862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
49	48	elfvexd 6914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝐷 ∈ V)
50	15, 17, 6, 49	catpropd 17797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
51	6, 50	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝐷 ∈ Cat)
52	42	simplrd 782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
53	52, 47	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
54		eqid 2760	. . . . . 6 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
55	44, 45, 51, 48, 53, 54	invfval 17848	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (((1^st ‘(1^st ‘𝑃))(Sect‘𝐷)(2^nd ‘(1^st ‘𝑃))) ∩ ^◡((2^nd ‘(1^st ‘𝑃))(Sect‘𝐷)(1^st ‘(1^st ‘𝑃)))))
56	22, 43, 55	3eqtr4rd 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃))
57		invfn 49956	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Inv‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
58	51, 57	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (Inv‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
59		fnbrovb 7464	. . . . 5 ⊢ (((Inv‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) ∧ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐷) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))) → (((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃) ↔ ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Inv‘𝐷)(2^nd ‘𝑃)))
60	58, 48, 53, 59	syl12anc 850	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → (((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃) ↔ ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Inv‘𝐷)(2^nd ‘𝑃)))
61	56, 60	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Inv‘𝐷)(2^nd ‘𝑃))
62		df-br 5104	. . 3 ⊢ (⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Inv‘𝐷)(2^nd ‘𝑃) ↔ ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩ ∈ (Inv‘𝐷))
63	61, 62	sylib 221	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩ ∈ (Inv‘𝐷))
64	13, 63	eqeltrd 2860	1 ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ∩ cin 3898 ⟨cop 4590 class class class wbr 5103 × cxp 5653 ^◡ccnv 5654 Fn wfn 6528 ‘cfv 6533 (class class class)co 7413 {coprab 7414 ∈ cmpo 7415 1^st c1st 7984 2^nd c2nd 7985 Basecbs 17301 Catccat 17752 Hom_f chomf 17754 comp_fccomf 17755 Sectcsect 17833 Invcinv 17834

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-cat 17756 df-cid 17757 df-homf 17758 df-comf 17759 df-sect 17836 df-inv 17837

This theorem is used by: invpropd 49965

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