Theorem fthcomf 49854

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 2769	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2769	. . . . . . 7 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
4		eqid 2769	. . . . . . 7 ⊢ (comp‘𝐴) = (comp‘𝐴)
5		eqid 2769	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
6		fthcomf.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)
7	6	ad2antrr 738	. . . . . . . 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 17966	. . . . . . . . 9 ⊢ (𝐴 Faith 𝐶) ⊆ (𝐴 Func 𝐶)
9	8	ssbri 5160	. . . . . . . 8 ⊢ (𝐹(𝐴 Faith 𝐶)𝐺 → 𝐹(𝐴 Func 𝐶)𝐺)
10	7, 9	syl 18	. . . . . . 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 1232	. . . . . . 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 1233	. . . . . . 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 1234	. . . . . . 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 782	. . . . . . 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 784	. . . . . . 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 17928	. . . . . 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 2769	. . . . . . 7 ⊢ (Base‘𝐵) = (Base‘𝐵)
18		eqid 2769	. . . . . . 7 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
19		eqid 2769	. . . . . . 7 ⊢ (comp‘𝐵) = (comp‘𝐵)
20		eqid 2769	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
21		fthcomf.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
22	21	ad2antrr 738	. . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
24	23, 21	funchomf 49794	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
25	24	homfeqbas 17752	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
26	25	ad2antrr 738	. . . . . . . 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 2871	. . . . . . 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 2871	. . . . . . 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 2871	. . . . . . 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 738	. . . . . . . . 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 17753	. . . . . . . 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 2871	. . . . . . 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 17753	. . . . . . . 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 2871	. . . . . . 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 17928	. . . . . 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 2814	. . . . 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 2769	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38	23	funcrcl2 49776	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Cat)
39	38	ad2antrr 738	. . . . . . 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 17741	. . . . . 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 49776	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Cat)
42	41	ad2antrr 738	. . . . . . . 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 17741	. . . . . . 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 17753	. . . . . . 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 2872	. . . . . 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 17977	. . . . 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 235	. . . 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 3214	. . 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 3217	. 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 2770	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
51	4, 19, 3, 50, 25, 24	comfeq 17762	. 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 260	1 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Hom chom 17321  compcco 17322  Catccat 17720  Hom_f chomf 17722  comp_fccomf 17723   Func cfunc 17911   Faith cfth 17962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-ixp 8896  df-cat 17724  df-homf 17726  df-comf 17727  df-func 17915  df-fth 17964

This theorem is referenced by: (None)