Theorem fthcomf 49045

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 2735	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2735	. . . . . . 7 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
4		eqid 2735	. . . . . . 7 ⊢ (comp‘𝐴) = (comp‘𝐴)
5		eqid 2735	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
6		fthcomf.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)
7	6	ad2antrr 726	. . . . . . . 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 17920	. . . . . . . . 9 ⊢ (𝐴 Faith 𝐶) ⊆ (𝐴 Func 𝐶)
9	8	ssbri 5164	. . . . . . . 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 1216	. . . . . . 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 1217	. . . . . . 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 1218	. . . . . . 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 770	. . . . . . 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 772	. . . . . . 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 17882	. . . . . 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 2735	. . . . . . 7 ⊢ (Base‘𝐵) = (Base‘𝐵)
18		eqid 2735	. . . . . . 7 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
19		eqid 2735	. . . . . . 7 ⊢ (comp‘𝐵) = (comp‘𝐵)
20		eqid 2735	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
21		fthcomf.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
22	21	ad2antrr 726	. . . . . . 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 49005	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
25	24	homfeqbas 17706	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
26	25	ad2antrr 726	. . . . . . . 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 2836	. . . . . . 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 2836	. . . . . . 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 2836	. . . . . . 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 726	. . . . . . . . 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 17707	. . . . . . . 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 2836	. . . . . . 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 17707	. . . . . . . 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 2836	. . . . . . 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 17882	. . . . . 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 2780	. . . . 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 2735	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38	23	funcrcl2 48992	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Cat)
39	38	ad2antrr 726	. . . . . . 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 17695	. . . . . 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 48992	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Cat)
42	41	ad2antrr 726	. . . . . . . 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 17695	. . . . . . 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 17707	. . . . . . 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 2837	. . . . . 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 17931	. . . . 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 3187	. . 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 3190	. 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 2736	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
51	4, 19, 3, 50, 25, 24	comfeq 17716	. 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 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ⟨cop 4607   class class class wbr 5119  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  Hom chom 17280  compcco 17281  Catccat 17674  Hom_f chomf 17676  comp_fccomf 17677   Func cfunc 17865   Faith cfth 17916

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-map 8840  df-ixp 8910  df-cat 17678  df-homf 17680  df-comf 17681  df-func 17869  df-fth 17918

This theorem is referenced by: (None)