Theorem metakunt12 40588

Description: C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)

Hypotheses

Ref	Expression
metakunt12.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt12.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt12.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt12.4	⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
metakunt12.5	⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
metakunt12.6	⊢ (𝜑 → 𝑋 ∈ (1...𝑀))

Assertion

Ref	Expression
metakunt12	⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑦,𝐼   𝑥,𝑀   𝑦,𝑀   𝑥,𝑋   𝑦,𝑋   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑦)

Proof of Theorem metakunt12

Step	Hyp	Ref	Expression
1		ioran 982	. 2 ⊢ (¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼) ↔ (¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼))
2		metakunt12.4	. . . . 5 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))))
4		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
5		breq1 5108	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
6		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
7		oveq1 7364	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
8	5, 6, 7	ifbieq12d 4514	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
9	4, 8	ifbieq2d 4512	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
10	9	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
11		metakunt12.5	. . . . . . . . . 10 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
12	11	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))))
13		eqeq1 2740	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
14		breq1 5108	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
15		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
16		oveq1 7364	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
17	14, 15, 16	ifbieq12d 4514	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
18	13, 17	ifbieq2d 4512	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
19	18	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
20		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
21		iffalse 4495	. . . . . . . . . . . . 13 ⊢ (¬ 𝑋 = 𝑀 → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
23		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼)
24		iffalse 4495	. . . . . . . . . . . . 13 ⊢ (¬ 𝑋 < 𝐼 → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑋 + 1))
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑋 + 1))
26	22, 25	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
27	26	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
28	19, 27	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = (𝑋 + 1))
29		metakunt12.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
30	29	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
31	29	elfzelzd 13442	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℤ)
32	31	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℤ)
33	32	peano2zd 12610	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ ℤ)
34	12, 28, 30, 33	fvmptd 6955	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) = (𝑋 + 1))
35		eqeq1 2740	. . . . . . . . 9 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) = 𝐼 ↔ (𝑋 + 1) = 𝐼))
36		breq1 5108	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) < 𝐼 ↔ (𝑋 + 1) < 𝐼))
37		id 22	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → (𝐶‘𝑋) = (𝑋 + 1))
38		oveq1 7364	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) − 1) = ((𝑋 + 1) − 1))
39	36, 37, 38	ifbieq12d 4514	. . . . . . . . 9 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
40	35, 39	ifbieq2d 4512	. . . . . . . 8 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))))
41	34, 40	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))))
42		metakunt12.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℕ)
43	42	nnred 12168	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ ℝ)
44	43	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
45	32	zred 12607	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
46	33	zred 12607	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ ℝ)
47	44, 45	lenltd 11301	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐼 ≤ 𝑋 ↔ ¬ 𝑋 < 𝐼))
48	23, 47	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑋)
49	45	ltp1d 12085	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 < (𝑋 + 1))
50	44, 45, 46, 48, 49	lelttrd 11313	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 < (𝑋 + 1))
51	44, 50	ltned 11291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≠ (𝑋 + 1))
52	51	necomd 2999	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ≠ 𝐼)
53	52	neneqd 2948	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ (𝑋 + 1) = 𝐼)
54		iffalse 4495	. . . . . . . . 9 ⊢ (¬ (𝑋 + 1) = 𝐼 → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
56	45	lep1d 12086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ≤ (𝑋 + 1))
57	44, 45, 46, 48, 56	letrd 11312	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ (𝑋 + 1))
58	44, 46	lenltd 11301	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐼 ≤ (𝑋 + 1) ↔ ¬ (𝑋 + 1) < 𝐼))
59	57, 58	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ (𝑋 + 1) < 𝐼)
60		iffalse 4495	. . . . . . . . 9 ⊢ (¬ (𝑋 + 1) < 𝐼 → if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)) = ((𝑋 + 1) − 1))
61	59, 60	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)) = ((𝑋 + 1) − 1))
62	32	zcnd 12608	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℂ)
63		1cnd 11150	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 1 ∈ ℂ)
64	62, 63	pncand 11513	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ((𝑋 + 1) − 1) = 𝑋)
65	55, 61, 64	3eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = 𝑋)
66	41, 65	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
67	66	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
68	10, 67	eqtrd 2776	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
69		metakunt12.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
70		metakunt12.3	. . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
71	69, 42, 70, 11	metakunt2 40578	. . . . . 6 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
72	71	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
73	72, 30	ffvelcdmd 7036	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
74	3, 68, 73, 30	fvmptd 6955	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
75	74	3expb 1120	. 2 ⊢ ((𝜑 ∧ (¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
76	1, 75	sylan2b 594	1 ⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  1c1 11052   + caddc 11054   < clt 11189   ≤ cle 11190   − cmin 11385  ℕcn 12153  ℤcz 12499  ...cfz 13424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425

This theorem is referenced by:  metakunt13  40589