Theorem monpropd 17704

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 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2736	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		monpropd.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
5	4	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 727	. . . . . . . . . . . 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 17663	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2736	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
12	4	ad5antr 735	. . . . . . . . . . . 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 735	. . . . . . . . . . . 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 17674	. . . . . . . . . . 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 5171	. . . . . . . . . 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 5830	. . . . . . . . 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 6520	. . . . . . . 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 3158	. . . . . . 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 3395	. . . . . 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 17663	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
29	4	homfeqbas 17662	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	29	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
31	30	raleqdv 3295	. . . . . . 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 3407	. . . . . 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 2771	. . . . 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 7444	. . 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 7440	. . . 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 585	. . 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 2771	. 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 2736	. . 3 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
40		monpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	1, 2, 10, 39, 40	monfval 17699	. 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 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		eqid 2736	. . 3 ⊢ (Mono‘𝐷) = (Mono‘𝐷)
44		monpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
45	42, 3, 11, 43, 44	monfval 17699	. 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 2781	1 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  ⟨cop 4573   ↦ cmpt 5166  ^◡ccnv 5630  Fun wfun 6492  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17630  Hom_f chomf 17632  comp_fccomf 17633  Monocmon 17695

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-homf 17636  df-comf 17637  df-mon 17697

This theorem is referenced by:  oppcepi  17706