Theorem monpropd 17001

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 2821	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2821	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2821	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		monpropd.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
5	4	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		simpr 487	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → 𝑐 ∈ (Base‘𝐶))
8		simp-4r 782	. . . . . . . . . . . 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 16961	. . . . . . . . . . 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 2821	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2821	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
12	4	ad5antr 732	. . . . . . . . . . . 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 732	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 784	. . . . . . . . . . . 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 782	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 774	. . . . . . . . . . . 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 16972	. . . . . . . . . . 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 5143	. . . . . . . . . 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 5741	. . . . . . . . 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 6372	. . . . . . . 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 3196	. . . . . . 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 3479	. . . . . 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 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑎 ∈ (Base‘𝐶))
27		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑏 ∈ (Base‘𝐶))
28	1, 2, 3, 5, 26, 27	homfeqval 16961	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
29	4	homfeqbas 16960	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	29	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
31	30	raleqdv 3416	. . . . . . 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 3486	. . . . . 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 2856	. . . . 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 1106	. . . 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 7225	. . 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 7221	. . . 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 586	. . 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 2856	. 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 2821	. . 3 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
40		monpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	1, 2, 10, 39, 40	monfval 16996	. 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 2821	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		eqid 2821	. . 3 ⊢ (Mono‘𝐷) = (Mono‘𝐷)
44		monpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
45	42, 3, 11, 43, 44	monfval 16996	. 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 2866	1 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  ⟨cop 4567   ↦ cmpt 5139  ^◡ccnv 5549  Fun wfun 6344  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  Hom_f chomf 16931  comp_fccomf 16932  Monocmon 16992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-homf 16935  df-comf 16936  df-mon 16994

This theorem is referenced by:  oppcepi  17003