Theorem imasubc 49772

Description: An image of a full functor is a full subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imasubc.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imasubc.h	⊢ 𝐻 = (Hom ‘𝐷)
imasubc.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
imasubc.f	⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)
imasubc.c	⊢ 𝐶 = (Base‘𝐸)
imasubc.j	⊢ 𝐽 = (Homf ‘𝐸)

Assertion

Ref	Expression
imasubc	⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ 𝐶 ∧ (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝐸,𝑝   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥,𝑦,𝑝)   𝐶(𝑥,𝑦,𝑝)   𝐷(𝑥,𝑦,𝑝)   𝑆(𝑝)   𝐸(𝑥,𝑦)   𝐽(𝑥,𝑦,𝑝)   𝐾(𝑥,𝑦,𝑝)

Proof of Theorem imasubc

Dummy variables 𝑚 𝑛 𝑞 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasubc.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)
2		relfull 17943	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3	2	brrelex1i 5703	. . . 4 ⊢ (𝐹(𝐷 Full 𝐸)𝐺 → 𝐹 ∈ V)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		imasubc.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
6	4, 4, 5	imasubclem2 49726	. 2 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
7		imasubc.s	. . 3 ⊢ 𝑆 = (𝐹 “ 𝐴)
8		eqid 2762	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		imasubc.c	. . . . 5 ⊢ 𝐶 = (Base‘𝐸)
10		fullfunc 17941	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
11	10	ssbri 5145	. . . . . 6 ⊢ (𝐹(𝐷 Full 𝐸)𝐺 → 𝐹(𝐷 Func 𝐸)𝐺)
12	1, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
13	8, 9, 12	funcf1 17899	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶𝐶)
14	13	fimassd 6713	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐶)
15	7, 14	eqsstrid 3974	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐶)
16		simprl 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
17	16, 7	eleqtrdi 2872	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ (𝐹 “ 𝐴))
18		inisegn0a 49457	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑧}) ≠ ∅)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡𝐹 “ {𝑧}) ≠ ∅)
20		simprr 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ 𝑆)
21	20, 7	eleqtrdi 2872	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ (𝐹 “ 𝐴))
22		inisegn0a 49457	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡𝐹 “ {𝑤}) ≠ ∅)
24	19, 23	jca 519	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡𝐹 “ {𝑧}) ≠ ∅ ∧ (◡𝐹 “ {𝑤}) ≠ ∅))
25		xpnz 6144	. . . . . . . 8 ⊢ (((◡𝐹 “ {𝑧}) ≠ ∅ ∧ (◡𝐹 “ {𝑤}) ≠ ∅) ↔ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤})) ≠ ∅)
26	24, 25	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤})) ≠ ∅)
27	13	ffnd 6692	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐷))
28	27	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝐹 Fn (Base‘𝐷))
29		simprl 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑚 ∈ (◡𝐹 “ {𝑧}))
30		fniniseg 7041	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑧}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧)))
31	30	biimpa 480	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn (Base‘𝐷) ∧ 𝑚 ∈ (◡𝐹 “ {𝑧})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧))
32	28, 29, 31	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧))
33	32	simprd 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝐹‘𝑚) = 𝑧)
34		simprr 782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑛 ∈ (◡𝐹 “ {𝑤}))
35		fniniseg 7041	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑛 ∈ (◡𝐹 “ {𝑤}) ↔ (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤)))
36	35	biimpa 480	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn (Base‘𝐷) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤})) → (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤))
37	28, 34, 36	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤))
38	37	simprd 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝐹‘𝑛) = 𝑤)
39	33, 38	oveq12d 7414	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → ((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
40		eqid 2762	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
41		imasubc.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (Hom ‘𝐷)
42	1	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝐹(𝐷 Full 𝐸)𝐺)
43	32	simpld 498	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑚 ∈ (Base‘𝐷))
44	37	simpld 498	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑛 ∈ (Base‘𝐷))
45	8, 40, 41, 42, 43, 44	fullfo 17947	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)))
46		foeq3 6776	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)) = (𝑧(Hom ‘𝐸)𝑤) → ((𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)) ↔ (𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→(𝑧(Hom ‘𝐸)𝑤)))
47	46	biimpa 480	. . . . . . . . . . 11 ⊢ ((((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)) = (𝑧(Hom ‘𝐸)𝑤) ∧ (𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛))) → (𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→(𝑧(Hom ‘𝐸)𝑤))
48	39, 45, 47	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→(𝑧(Hom ‘𝐸)𝑤))
49		foima 6783	. . . . . . . . . 10 ⊢ ((𝑚𝐺𝑛):(𝑚𝐻𝑛)–onto→(𝑧(Hom ‘𝐸)𝑤) → ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
51	50	ralrimivva 3205	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ∀𝑚 ∈ (◡𝐹 “ {𝑧})∀𝑛 ∈ (◡𝐹 “ {𝑤})((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
52		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑛⟩))
53		df-ov 7399	. . . . . . . . . . . 12 ⊢ (𝑚𝐺𝑛) = (𝐺‘⟨𝑚, 𝑛⟩)
54	52, 53	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑛))
55		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑛⟩))
56		df-ov 7399	. . . . . . . . . . . 12 ⊢ (𝑚𝐻𝑛) = (𝐻‘⟨𝑚, 𝑛⟩)
57	55, 56	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑛))
58	54, 57	imaeq12d 6050	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)))
59	58	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤) ↔ ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) = (𝑧(Hom ‘𝐸)𝑤)))
60	59	ralxp 5813	. . . . . . . 8 ⊢ (∀𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤) ↔ ∀𝑚 ∈ (◡𝐹 “ {𝑧})∀𝑛 ∈ (◡𝐹 “ {𝑤})((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
61	51, 60	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ∀𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤))
62		iuneqconst2 49444	. . . . . . 7 ⊢ ((((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤})) ≠ ∅ ∧ ∀𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤)) → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤))
63	26, 61, 62	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = (𝑧(Hom ‘𝐸)𝑤))
64	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐹 ∈ V)
65	64, 64, 16, 20, 5	imasubclem3 49727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐾𝑤) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
66		imasubc.j	. . . . . . 7 ⊢ 𝐽 = (Homf ‘𝐸)
67	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑆 ⊆ 𝐶)
68	67, 16	sseldd 3937	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ 𝐶)
69	67, 20	sseldd 3937	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ 𝐶)
70	66, 9, 40, 68, 69	homfval 17724	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧(Hom ‘𝐸)𝑤))
71	63, 65, 70	3eqtr4rd 2808	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
72	71	ralrimivva 3205	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
73		fveq2 6867	. . . . . . 7 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝐽‘⟨𝑧, 𝑤⟩))
74		df-ov 7399	. . . . . . 7 ⊢ (𝑧𝐽𝑤) = (𝐽‘⟨𝑧, 𝑤⟩)
75	73, 74	eqtr4di 2815	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝑧𝐽𝑤))
76		fveq2 6867	. . . . . . 7 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝐾‘⟨𝑧, 𝑤⟩))
77		df-ov 7399	. . . . . . 7 ⊢ (𝑧𝐾𝑤) = (𝐾‘⟨𝑧, 𝑤⟩)
78	76, 77	eqtr4di 2815	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝑧𝐾𝑤))
79	75, 78	eqeq12d 2778	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝐽‘𝑞) = (𝐾‘𝑞) ↔ (𝑧𝐽𝑤) = (𝑧𝐾𝑤)))
80	79	ralxp 5813	. . . 4 ⊢ (∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
81	72, 80	sylibr 236	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞))
82	66, 9	homffn 17725	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
83	82	a1i 11	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝐶 × 𝐶))
84		xpss12 5662	. . . . 5 ⊢ ((𝑆 ⊆ 𝐶 ∧ 𝑆 ⊆ 𝐶) → (𝑆 × 𝑆) ⊆ (𝐶 × 𝐶))
85	15, 15, 84	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐶 × 𝐶))
86		fvreseq1 7020	. . . 4 ⊢ (((𝐽 Fn (𝐶 × 𝐶) ∧ 𝐾 Fn (𝑆 × 𝑆)) ∧ (𝑆 × 𝑆) ⊆ (𝐶 × 𝐶)) → ((𝐽 ↾ (𝑆 × 𝑆)) = 𝐾 ↔ ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞)))
87	83, 6, 85, 86	syl21anc 848	. . 3 ⊢ (𝜑 → ((𝐽 ↾ (𝑆 × 𝑆)) = 𝐾 ↔ ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞)))
88	81, 87	mpbird 259	. 2 ⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)
89	6, 15, 88	3jca 1141	1 ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ 𝐶 ∧ (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  {csn 4582  ⟨cop 4588  ∪ ciun 4949   class class class wbr 5100   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650   Fn wfn 6516  –onto→wfo 6519  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  Basecbs 17245  Hom chom 17297  Hom_f chomf 17698   Func cfunc 17887   Full cful 17937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ixp 8880  df-homf 17702  df-func 17891  df-full 17939

This theorem is referenced by:  imasubc2  49773  idfullsubc  49782