Theorem fthcomf 49545

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 2737	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2737	. . . . . . 7 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
4		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐴) = (comp‘𝐴)
5		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
6		fthcomf.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)
7	6	ad2antrr 727	. . . . . . . 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 17847	. . . . . . . . 9 ⊢ (𝐴 Faith 𝐶) ⊆ (𝐴 Func 𝐶)
9	8	ssbri 5145	. . . . . . . 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 1217	. . . . . . 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 1218	. . . . . . 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 1219	. . . . . . 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 771	. . . . . . 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 773	. . . . . . 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 17809	. . . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝐵) = (Base‘𝐵)
18		eqid 2737	. . . . . . 7 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
19		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐵) = (comp‘𝐵)
20		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
21		fthcomf.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
22	21	ad2antrr 727	. . . . . . 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 49485	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
25	24	homfeqbas 17633	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
26	25	ad2antrr 727	. . . . . . . 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 2839	. . . . . . 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 2839	. . . . . . 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 2839	. . . . . . 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 727	. . . . . . . . 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 17634	. . . . . . . 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 2839	. . . . . . 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 17634	. . . . . . . 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 2839	. . . . . . 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 17809	. . . . . 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 2782	. . . . 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 2737	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38	23	funcrcl2 49467	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Cat)
39	38	ad2antrr 727	. . . . . . 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 17622	. . . . . 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 49467	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Cat)
42	41	ad2antrr 727	. . . . . . . 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 17622	. . . . . . 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 17634	. . . . . . 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 2840	. . . . . 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 17858	. . . . 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 232	. . . 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 3181	. . 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 3184	. 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 2738	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
51	4, 19, 3, 50, 25, 24	comfeq 17643	. 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 257	1 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4588   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  Hom chom 17202  compcco 17203  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604   Func cfunc 17792   Faith cfth 17843

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 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-map 8779  df-ixp 8850  df-cat 17605  df-homf 17607  df-comf 17608  df-func 17796  df-fth 17845

This theorem is referenced by: (None)