Theorem mapdhval 42984

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
mapdh.q	⊢ 𝑄 = (0g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))
mapdh.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
mapdh.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
mapdh.y	⊢ (𝜑 → 𝑌 ∈ 𝐸)

Assertion

Ref	Expression
mapdhval	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))

Distinct variable groups:   𝑥,𝐷   𝑥,ℎ,𝐹   𝑥,𝐽   𝑥,𝑀   𝑥,𝑁   𝑥, 0   𝑥,𝑄   𝑥,𝑅   𝑥, −   ℎ,𝑋,𝑥   ℎ,𝑌,𝑥   𝜑,ℎ

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥, ℎ)   𝐵(𝑥, ℎ)   𝐶(𝑥, ℎ)   𝐷(ℎ)   𝑄(ℎ)   𝑅(ℎ)   𝐸(𝑥, ℎ)   𝐼(𝑥, ℎ)   𝐽(ℎ)   𝑀(ℎ)   − (ℎ)   𝑁(ℎ)   0 (ℎ)

Proof of Theorem mapdhval

Step	Hyp	Ref	Expression
1		otex 5439	. . 3 ⊢ ⟨𝑋, 𝐹, 𝑌⟩ ∈ V
2		fveq2 6911	. . . . . 6 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))
3	2	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘𝑥) = 0 ↔ (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ))
4	2	sneqd 4596	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {(2nd ‘𝑥)} = {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})
5	4	fveq2d 6915	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{(2nd ‘𝑥)}) = (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}))
6	5	fveqeq2d 6919	. . . . . . 7 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ})))
7		fveq2 6911	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘𝑥) = (1st ‘⟨𝑋, 𝐹, 𝑌⟩))
8	7	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
9	8, 2	oveq12d 7464	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥)) = ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)))
10	9	sneqd 4596	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))} = {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})
11	10	fveq2d 6915	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))}) = (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}))
12	11	fveq2d 6915	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})))
13	7	fveq2d 6915	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
14	13	oveq1d 7461	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘(1st ‘𝑥))𝑅ℎ) = ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ))
15	14	sneqd 4596	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((2nd ‘(1st ‘𝑥))𝑅ℎ)} = {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})
16	15	fveq2d 6915	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))
17	12, 16	eqeq12d 2779	. . . . . . 7 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))
18	6, 17	anbi12d 644	. . . . . 6 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))))
19	18	riotabidv 7405	. . . . 5 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))))
20	3, 19	ifbieq2d 4512	. . . 4 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))))
21		mapdh.i	. . . 4 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))
22		mapdh.q	. . . . . 6 ⊢ 𝑄 = (0g‘𝐶)
23	22	fvexi 6925	. . . . 5 ⊢ 𝑄 ∈ V
24		riotaex 7407	. . . . 5 ⊢ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))) ∈ V
25	23, 24	ifex 4536	. . . 4 ⊢ if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))) ∈ V
26	20, 21, 25	fvmpt 7019	. . 3 ⊢ (⟨𝑋, 𝐹, 𝑌⟩ ∈ V → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))))
27	1, 26	mp1i 14	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))))
28		mapdh.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
29		ot3rdg 8044	. . . . 5 ⊢ (𝑌 ∈ 𝐸 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
30	28, 29	syl 18	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
31	30	eqeq1d 2765	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ↔ 𝑌 = 0 ))
32	30	sneqd 4596	. . . . . . 7 ⊢ (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
33	32	fveq2d 6915	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
34	33	fveqeq2d 6919	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ})))
35		mapdh.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
36		mapdh.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37		ot1stg 8042	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
38	35, 36, 28, 37	syl3anc 1399	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
39	38, 30	oveq12d 7464	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 − 𝑌))
40	39	sneqd 4596	. . . . . . . 8 ⊢ (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 − 𝑌)})
41	40	fveq2d 6915	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 − 𝑌)}))
42	41	fveq2d 6915	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
43		ot2ndg 8043	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
44	35, 36, 28, 43	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
45	44	oveq1d 7461	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ) = (𝐹𝑅ℎ))
46	45	sneqd 4596	. . . . . . 7 ⊢ (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)} = {(𝐹𝑅ℎ)})
47	46	fveq2d 6915	. . . . . 6 ⊢ (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) = (𝐽‘{(𝐹𝑅ℎ)}))
48	42, 47	eqeq12d 2779	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))
49	34, 48	anbi12d 644	. . . 4 ⊢ (𝜑 → (((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
50	49	riotabidv 7405	. . 3 ⊢ (𝜑 → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
51	31, 50	ifbieq2d 4512	. 2 ⊢ (𝜑 → if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))
52	27, 51	eqtrd 2798	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485  {csn 4588  ⟨cotp 4861   ↦ cmpt 5190  ‘cfv 6561  ℩crio 7402  (class class class)co 7446  1^st c1st 8026  2^nd c2nd 8027  0_gc0g 17642

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-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

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-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fv 6569  df-riota 7403  df-ov 7449  df-1st 8028  df-2nd 8029

This theorem is used by:  mapdhval0  42985  mapdhval2  42986  hdmap1valc  43063