Theorem erdsze2lem1 30439

Description: Lemma for erdsze2 30441. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypotheses

Ref	Expression
erdsze2.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdsze2.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdsze2.f	⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)
erdsze2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
erdsze2lem.n	⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
erdsze2lem.l	⊢ (𝜑 → 𝑁 < (#‘𝐴))

Assertion

Ref	Expression
erdsze2lem1	⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐹   𝑅,𝑓   𝑆,𝑓   𝑓,𝑁   𝜑,𝑓

Proof of Theorem erdsze2lem1

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze2lem.n	. . . . . . . . 9 ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
2		erdsze2.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3		nnm1nn0 11211	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ0)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 − 1) ∈ ℕ0)
5		erdsze2.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ)
6		nnm1nn0 11211	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ0)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 − 1) ∈ ℕ0)
8	4, 7	nn0mulcld 11233	. . . . . . . . 9 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ0)
9	1, 8	syl5eqel 2692	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
10		peano2nn0 11210	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
11		hashfz1 12996	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ0 → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
12	9, 10, 11	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
14		erdsze2lem.l	. . . . . . . 8 ⊢ (𝜑 → 𝑁 < (#‘𝐴))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝑁 < (#‘𝐴))
16		hashcl 13009	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
17		nn0ltp1le 11312	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (𝑁 < (#‘𝐴) ↔ (𝑁 + 1) ≤ (#‘𝐴)))
18	9, 16, 17	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 < (#‘𝐴) ↔ (𝑁 + 1) ≤ (#‘𝐴)))
19	15, 18	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 + 1) ≤ (#‘𝐴))
20	13, 19	eqbrtrd 4605	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (#‘(1...(𝑁 + 1))) ≤ (#‘𝐴))
21		fzfid 12634	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
22		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
23		hashdom 13029	. . . . . 6 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((#‘(1...(𝑁 + 1))) ≤ (#‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
24	21, 22, 23	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → ((#‘(1...(𝑁 + 1))) ≤ (#‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
25	20, 24	mpbid 221	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
26		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
27		fzfid 12634	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
28		isinffi 8701	. . . . . 6 ⊢ ((¬ 𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
29	26, 27, 28	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
30		erdsze2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
31		reex 9906	. . . . . . . 8 ⊢ ℝ ∈ V
32		ssexg 4732	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
33	30, 31, 32	sylancl 693	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
34	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ∈ V)
35		brdomg 7851	. . . . . 6 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴))
36	34, 35	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴))
37	29, 36	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
38	25, 37	pm2.61dan 828	. . 3 ⊢ (𝜑 → (1...(𝑁 + 1)) ≼ 𝐴)
39		domeng 7855	. . . 4 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
40	33, 39	syl 17	. . 3 ⊢ (𝜑 → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
41	38, 40	mpbid 221	. 2 ⊢ (𝜑 → ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴))
42		simprr 792	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ 𝐴)
43	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝐴 ⊆ ℝ)
44	42, 43	sstrd 3578	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ ℝ)
45		ltso 9997	. . . . 5 ⊢ < Or ℝ
46		soss 4977	. . . . 5 ⊢ (𝑠 ⊆ ℝ → ( < Or ℝ → < Or 𝑠))
47	44, 45, 46	mpisyl 21	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → < Or 𝑠)
48		fzfid 12634	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ∈ Fin)
49		simprl 790	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ≈ 𝑠)
50		enfi 8061	. . . . . 6 ⊢ ((1...(𝑁 + 1)) ≈ 𝑠 → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
51	49, 50	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
52	48, 51	mpbid 221	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ∈ Fin)
53		fz1iso 13103	. . . 4 ⊢ (( < Or 𝑠 ∧ 𝑠 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
54	47, 52, 53	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
55		isof1o 6473	. . . . . . . . . 10 ⊢ (𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → 𝑓:(1...(#‘𝑠))–1-1-onto→𝑠)
56	55	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(#‘𝑠))–1-1-onto→𝑠)
57		hashen 12997	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑠 ∈ Fin) → ((#‘(1...(𝑁 + 1))) = (#‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
58	48, 52, 57	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((#‘(1...(𝑁 + 1))) = (#‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
59	49, 58	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (#‘(1...(𝑁 + 1))) = (#‘𝑠))
60	12	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
61	59, 60	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (#‘𝑠) = (𝑁 + 1))
62	61	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (#‘𝑠) = (𝑁 + 1))
63	62	oveq2d 6565	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (1...(#‘𝑠)) = (1...(𝑁 + 1)))
64		f1oeq2 6041	. . . . . . . . . 10 ⊢ ((1...(#‘𝑠)) = (1...(𝑁 + 1)) → (𝑓:(1...(#‘𝑠))–1-1-onto→𝑠 ↔ 𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠))
65	63, 64	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓:(1...(#‘𝑠))–1-1-onto→𝑠 ↔ 𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠))
66	56, 65	mpbid 221	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠)
67		f1of1 6049	. . . . . . . 8 ⊢ (𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠 → 𝑓:(1...(𝑁 + 1))–1-1→𝑠)
68	66, 67	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1→𝑠)
69		simplrr 797	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑠 ⊆ 𝐴)
70		f1ss 6019	. . . . . . 7 ⊢ ((𝑓:(1...(𝑁 + 1))–1-1→𝑠 ∧ 𝑠 ⊆ 𝐴) → 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
71	68, 69, 70	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
72		simpr 476	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
73		f1ofo 6057	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘𝑠))–1-1-onto→𝑠 → 𝑓:(1...(#‘𝑠))–onto→𝑠)
74		forn 6031	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘𝑠))–onto→𝑠 → ran 𝑓 = 𝑠)
75		isoeq5 6471	. . . . . . . . 9 ⊢ (ran 𝑓 = 𝑠 → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)))
76	56, 73, 74, 75	4syl 19	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)))
77	72, 76	mpbird 246	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓))
78		isoeq4 6470	. . . . . . . 8 ⊢ ((1...(#‘𝑠)) = (1...(𝑁 + 1)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
79	63, 78	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
80	77, 79	mpbid 221	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))
81	71, 80	jca 553	. . . . 5 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
82	81	ex 449	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → (𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
83	82	eximdv 1833	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
84	54, 83	mpd 15	. 2 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
85	41, 84	exlimddv 1850	1 ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   Or wor 4958  ran crn 5039  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ≈ cen 7838   ≼ cdom 7839  Fincfn 7841  ℝcr 9814  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ...cfz 12197  #chash 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980

This theorem is referenced by:  erdsze2  30441