Theorem fthcomf 49783

Description: Source categories of a faithful functor have the same base, hom-sets and composition operation if the composition is compatible in images of the functor. (Contributed by Zhi Wang, 10-Nov-2025.)

Hypotheses

Ref	Expression
fthcomf.1	⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)
fthcomf.2	⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
fthcomf.3	⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))

Assertion

Ref	Expression
fthcomf	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦,𝑧   𝐵,𝑓,𝑔,𝑥,𝑦,𝑧   𝜑,𝑓,𝑔,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem fthcomf

Step	Hyp	Ref	Expression
1		fthcomf.3	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))
2		eqid 2764	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2764	. . . . . . 7 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
4		eqid 2764	. . . . . . 7 ⊢ (comp‘𝐴) = (comp‘𝐴)
5		eqid 2764	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
6		fthcomf.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)
7	6	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝐹(𝐴 Faith 𝐶)𝐺)
8		fthfunc 17944	. . . . . . . . 9 ⊢ (𝐴 Faith 𝐶) ⊆ (𝐴 Func 𝐶)
9	8	ssbri 5147	. . . . . . . 8 ⊢ (𝐹(𝐴 Faith 𝐶)𝐺 → 𝐹(𝐴 Func 𝐶)𝐺)
10	7, 9	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝐹(𝐴 Func 𝐶)𝐺)
11		simplr1 1230	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑥 ∈ (Base‘𝐴))
12		simplr2 1231	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑦 ∈ (Base‘𝐴))
13		simplr3 1232	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑧 ∈ (Base‘𝐴))
14		simprl 780	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦))
15		simprr 782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))
16	2, 3, 4, 5, 10, 11, 12, 13, 14, 15	funcco 17906	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))
17		eqid 2764	. . . . . . 7 ⊢ (Base‘𝐵) = (Base‘𝐵)
18		eqid 2764	. . . . . . 7 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
19		eqid 2764	. . . . . . 7 ⊢ (comp‘𝐵) = (comp‘𝐵)
20		eqid 2764	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
21		fthcomf.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
22	21	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝐹(𝐵 Func 𝐷)𝐺)
23	6, 9	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
24	23, 21	funchomf 49723	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
25	24	homfeqbas 17730	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
26	25	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (Base‘𝐴) = (Base‘𝐵))
27	11, 26	eleqtrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑥 ∈ (Base‘𝐵))
28	12, 26	eleqtrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑦 ∈ (Base‘𝐵))
29	13, 26	eleqtrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑧 ∈ (Base‘𝐵))
30	24	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (Homf ‘𝐴) = (Homf ‘𝐵))
31	2, 3, 18, 30, 11, 12	homfeqval 17731	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
32	14, 31	eleqtrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐵)𝑦))
33	2, 3, 18, 30, 12, 13	homfeqval 17731	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑦(Hom ‘𝐴)𝑧) = (𝑦(Hom ‘𝐵)𝑧))
34	15, 33	eleqtrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐵)𝑧))
35	17, 18, 19, 20, 22, 27, 28, 29, 32, 34	funcco 17906	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))
36	1, 16, 35	3eqtr4d 2809	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓)) = ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓)))
37		eqid 2764	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38	23	funcrcl2 49705	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Cat)
39	38	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝐴 ∈ Cat)
40	2, 3, 4, 39, 11, 12, 13, 14, 15	catcocl 17719	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐴)𝑧))
41	21	funcrcl2 49705	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Cat)
42	41	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → 𝐵 ∈ Cat)
43	17, 18, 19, 42, 27, 28, 29, 32, 34	catcocl 17719	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐵)𝑧))
44	2, 3, 18, 30, 11, 13	homfeqval 17731	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑥(Hom ‘𝐴)𝑧) = (𝑥(Hom ‘𝐵)𝑧))
45	43, 44	eleqtrrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐴)𝑧))
46	2, 3, 37, 7, 11, 13, 40, 45	fthi 17955	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓)) = ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓)) ↔ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓)))
47	36, 46	mpbid 234	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓))
48	47	ralrimivva 3207	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓))
49	48	ralrimivvva 3210	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀𝑧 ∈ (Base‘𝐴)∀𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓))
50		eqidd 2765	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
51	4, 19, 3, 50, 25, 24	comfeq 17740	. 2 ⊢ (𝜑 → ((compf‘𝐴) = (compf‘𝐵) ↔ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀𝑧 ∈ (Base‘𝐴)∀𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐴)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐵)𝑧)𝑓)))
52	49, 51	mpbird 259	1 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ⟨cop 4590   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  Hom chom 17299  compcco 17300  Catccat 17698  Hom_f chomf 17700  comp_fccomf 17701   Func cfunc 17889   Faith cfth 17940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-ixp 8882  df-cat 17702  df-homf 17704  df-comf 17705  df-func 17893  df-fth 17942

This theorem is referenced by: (None)