Theorem monpropd 17781

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 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2737	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2737	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		monpropd.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
5	4	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑎(Hom ‘𝐶)𝑏)) ∧ 𝑐 ∈ (Base‘𝐶)) → 𝑐 ∈ (Base‘𝐶))
8		simp-4r 784	. . . . . . . . . . . 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 17740	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2737	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
12	4	ad5antr 734	. . . . . . . . . . . 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 734	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 786	. . . . . . . . . . . 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 784	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 776	. . . . . . . . . . . 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 17751	. . . . . . . . . . 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 5231	. . . . . . . . . 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 5886	. . . . . . . . 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 6588	. . . . . . . 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 3176	. . . . . . 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 3443	. . . . . 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 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑎 ∈ (Base‘𝐶))
27		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → 𝑏 ∈ (Base‘𝐶))
28	1, 2, 3, 5, 26, 27	homfeqval 17740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
29	4	homfeqbas 17739	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	29	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
31	30	raleqdv 3326	. . . . . . 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 3455	. . . . . 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 2777	. . . . 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 1110	. . . 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 7510	. . 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 7506	. . . 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 584	. . 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 2777	. 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 2737	. . 3 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
40		monpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	1, 2, 10, 39, 40	monfval 17776	. 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 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		eqid 2737	. . 3 ⊢ (Mono‘𝐷) = (Mono‘𝐷)
44		monpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
45	42, 3, 11, 43, 44	monfval 17776	. 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 2787	1 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  ⟨cop 4632   ↦ cmpt 5225  ^◡ccnv 5684  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Hom_f chomf 17709  comp_fccomf 17710  Monocmon 17772

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-homf 17713  df-comf 17714  df-mon 17774

This theorem is referenced by:  oppcepi  17783