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Theorem offval22 8134

Description: The function operation expressed as a mapping, variation of offval2 7745. (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 7797	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
4		xp1st 8067	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
5		xp2nd 8068	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
6	4, 5	jca 521	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵))
7		fvex 6929	. . . . . 6 ⊢ (2^nd ‘𝑧) ∈ V
8		fvex 6929	. . . . . 6 ⊢ (1^st ‘𝑧) ∈ V
9		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦(2^nd ‘𝑧)
10		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥(2^nd ‘𝑧)
11		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥(1^st ‘𝑧)
12		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
13		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐶
14	13	nfel1 2942	. . . . . . . 8 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
15	12, 14	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
16		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
17		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2942	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19	16, 18	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
20		eleq1 2852	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝑦 ∈ 𝐵 ↔ (2^nd ‘𝑧) ∈ 𝐵))
21	20	3anbi3d 1471	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)))
22		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → 𝐶 = ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
23	22	eleq1d 2849	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
24	21, 23	imbi12d 347	. . . . . . 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 2852	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (𝑥 ∈ 𝐴 ↔ (1^st ‘𝑧) ∈ 𝐴))
26	25	3anbi2d 1470	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ↔ (𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)))
27		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
28	27	eleq1d 2849	. . . . . . . 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 347	. . . . . . 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 3477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ V)
32	9, 10, 11, 15, 19, 24, 29, 31	vtocl2gf 3535	. . . . . 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 705	. . . . 5 ⊢ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
34	33	3expb 1138	. . . 4 ⊢ ((𝜑 ∧ ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
35	6, 34	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
36		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐷
37	36	nfel1 2942	. . . . . . . 8 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V
38	12, 37	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
39		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷
40	39	nfel1 2942	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V
41	16, 40	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
42		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → 𝐷 = ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
43	42	eleq1d 2849	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝐷 ∈ V ↔ ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V))
44	21, 43	imbi12d 347	. . . . . . 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 3865	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → ⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 = ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
46	45	eleq1d 2849	. . . . . . . 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 347	. . . . . . 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 3477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
50	9, 10, 11, 38, 41, 44, 47, 49	vtocl2gf 3535	. . . . . 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 705	. . . . 5 ⊢ ((𝜑 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
52	51	3expb 1138	. . . 4 ⊢ ((𝜑 ∧ ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)) → ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷 ∈ V)
53	6, 52	sylan2 605	. . 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 8111	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
56	54, 55	eqtrdi 2814	. . 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 8111	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷)
59	57, 58	eqtrdi 2814	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
60	3, 35, 53, 56, 59	offval2 7745	. 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 7497	. . . . . . 7 ⊢ ((2^nd ‘𝑧) ∈ V → ⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷) = (⦋(2^nd ‘𝑧) / 𝑦⦌𝐶𝑅⦋(2^nd ‘𝑧) / 𝑦⦌𝐷))
62	61	csbeq2dv 3858	. . . . . 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 7497	. . . . . 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 2787	. . . 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 5207	. . 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 8111	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(𝐶𝑅𝐷))
69	67, 68	eqtr4i 2789	. 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 2814	1 ⊢ (𝜑 → (𝐹 ∘_f 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⦋csb 3851 ↦ cmpt 5192 × cxp 5656 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 ∘_f cof 7723 1^st c1st 8033 2^nd c2nd 8034

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-1st 8035 df-2nd 8036

This theorem is used by: matsc 22820 mdetrsca2 22974 mdetrlin2 22977 mdetunilem5 22986 smadiadetglem2 23042 mat2pmatghm 23103 pm2mpghm 23189 fedgmullem1 34372 fedgmullem2 34373

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