Theorem heiborlem1 35093

Description: Lemma for heibor 35103. We work with a fixed open cover 𝑈 throughout. The set 𝐾 is the set of all subsets of 𝑋 that admit no finite subcover of 𝑈. (We wish to prove that 𝐾 is empty.) If a set 𝐶 has no finite subcover, then any finite cover of 𝐶 must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heiborlem1.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
heiborlem1	⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑢,𝑣,𝐷   𝑢,𝐵,𝑣   𝑢,𝐽,𝑣,𝑥   𝑢,𝑈,𝑣,𝑥   𝑢,𝐶,𝑣   𝑥,𝐾

Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐵(𝑥)   𝐶(𝑥)   𝐾(𝑣,𝑢)

Proof of Theorem heiborlem1

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		heiborlem1.4	. . . . . . . 8 ⊢ 𝐵 ∈ V
2		sseq1 3995	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ 𝑣))
3	2	rexbidv 3300	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣))
4	3	notbid 320	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣))
5		heibor.3	. . . . . . . 8 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
6	1, 4, 5	elab2 3673	. . . . . . 7 ⊢ (𝐵 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
7	6	con2bii 360	. . . . . 6 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ¬ 𝐵 ∈ 𝐾)
8	7	ralbii 3168	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾)
9		ralnex 3239	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)
10	8, 9	bitr2i 278	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
11		unieq 4852	. . . . . . . . 9 ⊢ (𝑣 = (𝑡‘𝑥) → ∪ 𝑣 = ∪ (𝑡‘𝑥))
12	11	sseq2d 4002	. . . . . . . 8 ⊢ (𝑣 = (𝑡‘𝑥) → (𝐵 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ (𝑡‘𝑥)))
13	12	ac6sfi 8765	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)))
14	13	ex 415	. . . . . 6 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))))
15	14	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))))
16		sseq1 3995	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐶 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ 𝑣))
17	16	rexbidv 3300	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
18	17	notbid 320	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
19	18, 5	elab2g 3671	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐾 → (𝐶 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
20	19	ibi 269	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
21		frn 6523	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
22	21	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23		inss1 4208	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
24	22, 23	sstrdi 3982	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25		sspwuni 5025	. . . . . . . . . . . . 13 ⊢ (ran 𝑡 ⊆ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
26	24, 25	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ⊆ 𝑈)
27		vex 3500	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
28	27	rnex 7620	. . . . . . . . . . . . . 14 ⊢ ran 𝑡 ∈ V
29	28	uniex 7470	. . . . . . . . . . . . 13 ⊢ ∪ ran 𝑡 ∈ V
30	29	elpw 4546	. . . . . . . . . . . 12 ⊢ (∪ ran 𝑡 ∈ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
31	26, 30	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ 𝒫 𝑈)
32		ffn 6517	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
33	32	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡 Fn 𝐴)
34		dffn4 6599	. . . . . . . . . . . . . 14 ⊢ (𝑡 Fn 𝐴 ↔ 𝑡:𝐴–onto→ran 𝑡)
35	33, 34	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡:𝐴–onto→ran 𝑡)
36		fofi 8813	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴–onto→ran 𝑡) → ran 𝑡 ∈ Fin)
37	35, 36	syldan 593	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ∈ Fin)
38		inss2 4209	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ Fin
39	22, 38	sstrdi 3982	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ Fin)
40		unifi 8816	. . . . . . . . . . . 12 ⊢ ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ∪ ran 𝑡 ∈ Fin)
41	37, 39, 40	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ Fin)
42	31, 41	elind 4174	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
43	42	adantlr 713	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44		simplr 767	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
45		fnfvelrn 6851	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
46	32, 45	sylan 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
47	46	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
48		elssuni 4871	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑥) ∈ ran 𝑡 → (𝑡‘𝑥) ⊆ ∪ ran 𝑡)
49		uniss 4849	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑥) ⊆ ∪ ran 𝑡 → ∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡)
50	47, 48, 49	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → ∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡)
51		sstr2 3977	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ⊆ ∪ (𝑡‘𝑥) → (∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡 → 𝐵 ⊆ ∪ ∪ ran 𝑡))
52	50, 51	syl5com 31	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ ∪ (𝑡‘𝑥) → 𝐵 ⊆ ∪ ∪ ran 𝑡))
53	52	ralimdva 3180	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡))
54	53	impr 457	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
55		iunss 4972	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
56	54, 55	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
57	56	adantlr 713	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
58	44, 57	sstrd 3980	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ ∪ ran 𝑡)
59		unieq 4852	. . . . . . . . . . 11 ⊢ (𝑣 = ∪ ran 𝑡 → ∪ 𝑣 = ∪ ∪ ran 𝑡)
60	59	sseq2d 4002	. . . . . . . . . 10 ⊢ (𝑣 = ∪ ran 𝑡 → (𝐶 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ ∪ ran 𝑡))
61	60	rspcev 3626	. . . . . . . . 9 ⊢ ((∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ⊆ ∪ ∪ ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
62	43, 58, 61	syl2anc 586	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
63	20, 62	nsyl3 140	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ¬ 𝐶 ∈ 𝐾)
64	63	ex 415	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)) → ¬ 𝐶 ∈ 𝐾))
65	64	exlimdv 1933	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)) → ¬ 𝐶 ∈ 𝐾))
66	15, 65	syld 47	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ¬ 𝐶 ∈ 𝐾))
67	10, 66	syl5bi 244	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾 → ¬ 𝐶 ∈ 𝐾))
68	67	con4d 115	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐶 ∈ 𝐾 → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾))
69	68	3impia 1113	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841  ∪ ciun 4922  ran crn 5559   Fn wfn 6353  ⟶wf 6354  –onto→wfo 6356  ‘cfv 6358  Fincfn 8512  MetOpencmopn 20538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8513  df-dom 8514  df-fin 8516

This theorem is referenced by:  heiborlem3  35095  heiborlem10  35102