Theorem monpropd 17832

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
monpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
monpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
monpropd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
monpropd.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
monpropd	⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))

Proof of Theorem monpropd

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2762	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2762	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		monpropd.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
5	4	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → 𝑐 ∈ (Base‘𝐶))
8		simp-4r 796	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → 𝑎 ∈ (Base‘𝐶))
9	1, 2, 3, 6, 7, 8	homfeqval 17791	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (𝑐(Hom ‘𝐶)𝑎) = (𝑐(Hom ‘𝐷)𝑎))
10		eqid 2762	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2762	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
12	4	ad5antr 747	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → (Homf ‘𝐶) = (Homf ‘𝐷))
13		monpropd.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
14	13	ad5antr 747	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → (compf‘𝐶) = (compf‘𝐷))
15		simplr 781	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → 𝑐 ∈ (Base‘𝐶))
16		simp-5r 798	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → 𝑎 ∈ (Base‘𝐶))
17		simp-4r 796	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → 𝑏 ∈ (Base‘𝐶))
18		simpr 490	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎))
19		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏))
20	1, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19	comfeqval 17802	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎)) → (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔) = (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))
21	9, 20	mpteq12dva 5195	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔)) = (𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔)))
22	21	cnveqd 5859	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔)) = ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔)))
23	22	funeqd 6559	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))))
24	23	ralbidva 3185	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) → (∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔)) ↔ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))))
25	24	rabbidva 3420	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))} = {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))})
26		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑎 ∈ (Base‘𝐶))
27		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑏 ∈ (Base‘𝐶))
28	1, 2, 3, 5, 26, 27	homfeqval 17791	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
29	4	homfeqbas 17790	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	29	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
31	30	raleqdv 3321	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔)) ↔ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))))
32	28, 31	rabeqbidv 3432	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))} = {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))})
33	25, 32	eqtrd 2797	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))} = {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))})
34	33	3impa 1127	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) → {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))} = {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))})
35	34	mpoeq3dva 7494	. . 3 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))}) = (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}))
36		mpoeq12 7490	. . . 4 ⊢ (((Base‘𝐶) = (Base‘𝐷) ∧ (Base‘𝐶) = (Base‘𝐷)) → (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}) = (𝑎 ∈ (Base‘𝐷), 𝑏 ∈ (Base‘𝐷) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}))
37	29, 29, 36	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}) = (𝑎 ∈ (Base‘𝐷), 𝑏 ∈ (Base‘𝐷) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}))
38	35, 37	eqtrd 2797	. 2 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))}) = (𝑎 ∈ (Base‘𝐷), 𝑏 ∈ (Base‘𝐷) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}))
39		eqid 2762	. . 3 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
40		monpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	1, 2, 10, 39, 40	monfval 17827	. 2 ⊢ (𝜑 → (Mono‘𝐶) = (𝑎 ∈ (Base‘𝐶), 𝑏 ∈ (Base‘𝐶) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐶)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐶)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐶)𝑏)𝑔))}))
42		eqid 2762	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		eqid 2762	. . 3 ⊢ (Mono‘𝐷) = (Mono‘𝐷)
44		monpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
45	42, 3, 11, 43, 44	monfval 17827	. 2 ⊢ (𝜑 → (Mono‘𝐷) = (𝑎 ∈ (Base‘𝐷), 𝑏 ∈ (Base‘𝐷) ↦ {𝑓 ∈ (𝑎(Hom ‘𝐷)𝑏) ∣ ∀𝑐 ∈ (Base‘𝐷)Fun ◡(𝑔 ∈ (𝑐(Hom ‘𝐷)𝑎) ↦ (𝑓(⟨𝑐, 𝑎⟩(comp‘𝐷)𝑏)𝑔))}))
46	38, 41, 45	3eqtr4d 2807	1 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  ⟨cop 4593   ↦ cmpt 5190  ^◡ccnv 5658  Fun wfun 6531  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Basecbs 17307  Hom chom 17359  compcco 17360  Catccat 17758  Hom_f chomf 17760  comp_fccomf 17761  Monocmon 17823

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-homf 17764  df-comf 17765  df-mon 17825

This theorem is used by:  oppcepi  17834