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Theorem offval22 8067

Description: The function operation expressed as a mapping, variation of offval2 7673. (Contributed by SO, 15-Jul-2018.)

**Hypotheses**
Ref	Expression
offval22.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval22.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval22.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋)
offval22.d	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌)
offval22.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
offval22.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))

**Assertion**
Ref	Expression
offval22	⊢ (𝜑 → (𝐹 ∘_f 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)))

Distinct variable groups: 𝜑,𝑥,𝑦 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝑅,𝑦 Allowed substitution hints: 𝐶(𝑥,𝑦) 𝐷(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝑉(𝑥,𝑦) 𝑊(𝑥,𝑦) 𝑋(𝑥,𝑦) 𝑌(𝑥,𝑦)

Proof of Theorem offval22 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		offval22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3	1, 2	xpexd 7727	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
4		xp1st 8000	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
5		xp2nd 8001	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
6	4, 5	jca 511	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵))
7		fvex 6871	. . . . . 6 ⊢ (2^nd ‘𝑧) ∈ V
8		fvex 6871	. . . . . 6 ⊢ (1^st ‘𝑧) ∈ V
9		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦(2^nd ‘𝑧)
10		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(2^nd ‘𝑧)
11		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(1^st ‘𝑧)
12		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
13		nfcsb1v 3886	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐶
14	13	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
15	12, 14	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
16		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
17		nfcsb1v 3886	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19	16, 18	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
20		eleq1 2816	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝑦 ∈ 𝐵 ↔ (2^nd ‘𝑧) ∈ 𝐵))
21	20	3anbi3d 1444	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)))
22		csbeq1a 3876	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → 𝐶 = ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
23	22	eleq1d 2813	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
24	21, 23	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = (2^nd ‘𝑧) → (((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ V) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)))
25		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (𝑥 ∈ 𝐴 ↔ (1^st ‘𝑧) ∈ 𝐴))
26	25	3anbi2d 1443	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ↔ (𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)))
27		csbeq1a 3876	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
28	27	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → (⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
29	26, 28	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V) ↔ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)))
30		offval22.c	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋)
31	30	elexd 3471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ V)
32	9, 10, 11, 15, 19, 24, 29, 31	vtocl2gf 3538	. . . . . 6 ⊢ (((2^nd ‘𝑧) ∈ V ∧ (1^st ‘𝑧) ∈ V) → ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
33	7, 8, 32	mp2an 692	. . . . 5 ⊢ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
34	33	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
35	6, 34	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
36		nfcsb1v 3886	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐷
37	36	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V
38	12, 37	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
39		nfcsb1v 3886	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷
40	39	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V
41	16, 40	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
42		csbeq1a 3876	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → 𝐷 = ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
43	42	eleq1d 2813	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝐷 ∈ V ↔ ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V))
44	21, 43	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = (2^nd ‘𝑧) → (((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)))
45		csbeq1a 3876	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 = ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
46	45	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → (⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V ↔ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V))
47	26, 46	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V) ↔ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)))
48		offval22.d	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌)
49	48	elexd 3471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
50	9, 10, 11, 38, 41, 44, 47, 49	vtocl2gf 3538	. . . . . 6 ⊢ (((2^nd ‘𝑧) ∈ V ∧ (1^st ‘𝑧) ∈ V) → ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V))
51	7, 8, 50	mp2an 692	. . . . 5 ⊢ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
52	51	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
53	6, 52	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
54		offval22.f	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
55		mpompts 8044	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
56	54, 55	eqtrdi 2780	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶))
57		offval22.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
58		mpompts 8044	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
59	57, 58	eqtrdi 2780	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
60	3, 35, 53, 56, 59	offval2 7673	. 2 ⊢ (𝜑 → (𝐹 ∘_f 𝑅𝐺) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)))
61		csbov12g 7433	. . . . . . 7 ⊢ ((2^nd ‘𝑧) ∈ V → ⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷) = (⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
62	61	csbeq2dv 3869	. . . . . 6 ⊢ ((2^nd ‘𝑧) ∈ V → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷) = ⦋(1^st ‘𝑧) / 𝑥⦌(⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
63	7, 62	ax-mp 5	. . . . 5 ⊢ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷) = ⦋(1^st ‘𝑧) / 𝑥⦌(⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
64		csbov12g 7433	. . . . . 6 ⊢ ((1^st ‘𝑧) ∈ V → ⦋(1^st ‘𝑧) / 𝑥⦌(⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷) = (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
65	8, 64	ax-mp 5	. . . . 5 ⊢ ⦋(1^st ‘𝑧) / 𝑥⦌(⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷) = (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
66	63, 65	eqtr2i 2753	. . . 4 ⊢ (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷) = ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷)
67	66	mpteq2i 5203	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷))
68		mpompts 8044	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷))
69	67, 68	eqtr4i 2755	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷))
70	60, 69	eqtrdi 2780	1 ⊢ (𝜑 → (𝐹 ∘_f 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ⦋csb 3862 ↦ cmpt 5188 × cxp 5636 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ∘_f cof 7651 1^st c1st 7966 2^nd c2nd 7967

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-1st 7968 df-2nd 7969

This theorem is referenced by: matsc 22337 mdetrsca2 22491 mdetrlin2 22494 mdetunilem5 22503 smadiadetglem2 22559 mat2pmatghm 22617 pm2mpghm 22703 fedgmullem1 33625 fedgmullem2 33626

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