Theorem mapdhval 37013

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 4933	. . 3 ⊢ ⟨𝑋, 𝐹, 𝑌⟩ ∈ V
2		fveq2 6191	. . . . . 6 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))
3	2	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘𝑥) = 0 ↔ (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ))
4	2	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {(2nd ‘𝑥)} = {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})
5	4	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{(2nd ‘𝑥)}) = (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}))
6	5	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})))
7	6	eqeq1d 2624	. . . . . . 7 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ})))
8		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘𝑥) = (1st ‘⟨𝑋, 𝐹, 𝑌⟩))
9	8	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
10	9, 2	oveq12d 6668	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥)) = ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)))
11	10	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))} = {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})
12	11	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))}) = (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}))
13	12	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})))
14	8	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
15	14	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘(1st ‘𝑥))𝑅ℎ) = ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ))
16	15	sneqd 4189	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((2nd ‘(1st ‘𝑥))𝑅ℎ)} = {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})
17	16	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))
18	13, 17	eqeq12d 2637	. . . . . . 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)})))
19	7, 18	anbi12d 747	. . . . . 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)}))))
20	19	riotabidv 6613	. . . . 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)}))))
21	3, 20	ifbieq2d 4111	. . . 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)})))))
22		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)})))))
23		mapdh.q	. . . . . 6 ⊢ 𝑄 = (0g‘𝐶)
24		fvex 6201	. . . . . 6 ⊢ (0g‘𝐶) ∈ V
25	23, 24	eqeltri 2697	. . . . 5 ⊢ 𝑄 ∈ V
26		riotaex 6615	. . . . 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
27	25, 26	ifex 4156	. . . 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
28	21, 22, 27	fvmpt 6282	. . 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)})))))
29	1, 28	mp1i 13	. 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)})))))
30		mapdh.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
31		ot3rdg 7184	. . . . 5 ⊢ (𝑌 ∈ 𝐸 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
32	30, 31	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
33	32	eqeq1d 2624	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ↔ 𝑌 = 0 ))
34	32	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
35	34	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
36	35	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝑀‘(𝑁‘{𝑌})))
37	36	eqeq1d 2624	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ})))
38		mapdh.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
39		mapdh.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
40		ot1stg 7182	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
41	38, 39, 30, 40	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
42	41, 32	oveq12d 6668	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 − 𝑌))
43	42	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 − 𝑌)})
44	43	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 − 𝑌)}))
45	44	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
46		ot2ndg 7183	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
47	38, 39, 30, 46	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
48	47	oveq1d 6665	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ) = (𝐹𝑅ℎ))
49	48	sneqd 4189	. . . . . . 7 ⊢ (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)} = {(𝐹𝑅ℎ)})
50	49	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) = (𝐽‘{(𝐹𝑅ℎ)}))
51	45, 50	eqeq12d 2637	. . . . 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)})))
52	37, 51	anbi12d 747	. . . 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)}))))
53	52	riotabidv 6613	. . 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)}))))
54	33, 53	ifbieq2d 4111	. 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)})))))
55	29, 54	eqtrd 2656	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ifcif 4086  {csn 4177  ⟨cotp 4185   ↦ cmpt 4729  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0_gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mapdhval0  37014  mapdhval2  37015  hdmap1valc  37093