Theorem heiborlem1 37811

Description: Lemma for heibor 37821. 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 3961	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ 𝑣))
3	2	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣))
4	3	notbid 318	. . . . . . . 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 3638	. . . . . . 7 ⊢ (𝐵 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
7	6	con2bii 357	. . . . . 6 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ¬ 𝐵 ∈ 𝐾)
8	7	ralbii 3075	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾)
9		ralnex 3055	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)
10	8, 9	bitr2i 276	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
11		unieq 4869	. . . . . . . . 9 ⊢ (𝑣 = (𝑡‘𝑥) → ∪ 𝑣 = ∪ (𝑡‘𝑥))
12	11	sseq2d 3968	. . . . . . . 8 ⊢ (𝑣 = (𝑡‘𝑥) → (𝐵 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ (𝑡‘𝑥)))
13	12	ac6sfi 9173	. . . . . . 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 412	. . . . . 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 480	. . . . 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 3961	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐶 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ 𝑣))
17	16	rexbidv 3153	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
18	17	notbid 318	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
19	18, 5	elab2g 3636	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐾 → (𝐶 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
20	19	ibi 267	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
21		frn 6659	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
22	21	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23		inss1 4188	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
24	22, 23	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25		sspwuni 5049	. . . . . . . . . . . . 13 ⊢ (ran 𝑡 ⊆ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
26	24, 25	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ⊆ 𝑈)
27		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
28	27	rnex 7843	. . . . . . . . . . . . . 14 ⊢ ran 𝑡 ∈ V
29	28	uniex 7677	. . . . . . . . . . . . 13 ⊢ ∪ ran 𝑡 ∈ V
30	29	elpw 4555	. . . . . . . . . . . 12 ⊢ (∪ ran 𝑡 ∈ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
31	26, 30	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ 𝒫 𝑈)
32		ffn 6652	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
33	32	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡 Fn 𝐴)
34		dffn4 6742	. . . . . . . . . . . . . 14 ⊢ (𝑡 Fn 𝐴 ↔ 𝑡:𝐴–onto→ran 𝑡)
35	33, 34	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡:𝐴–onto→ran 𝑡)
36		fofi 9202	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴–onto→ran 𝑡) → ran 𝑡 ∈ Fin)
37	35, 36	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ∈ Fin)
38		inss2 4189	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ Fin
39	22, 38	sstrdi 3948	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ Fin)
40		unifi 9234	. . . . . . . . . . . 12 ⊢ ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ∪ ran 𝑡 ∈ Fin)
41	37, 39, 40	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ Fin)
42	31, 41	elind 4151	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
43	42	adantlr 715	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
45		fnfvelrn 7014	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
46	32, 45	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
47	46	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
48		elssuni 4888	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑥) ∈ ran 𝑡 → (𝑡‘𝑥) ⊆ ∪ ran 𝑡)
49		uniss 4866	. . . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡))
54	53	impr 454	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
55		iunss 4994	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
56	54, 55	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
57	56	adantlr 715	. . . . . . . . . 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 3946	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ ∪ ran 𝑡)
59		unieq 4869	. . . . . . . . . . 11 ⊢ (𝑣 = ∪ ran 𝑡 → ∪ 𝑣 = ∪ ∪ ran 𝑡)
60	59	sseq2d 3968	. . . . . . . . . 10 ⊢ (𝑣 = ∪ ran 𝑡 → (𝐶 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ ∪ ran 𝑡))
61	60	rspcev 3577	. . . . . . . . 9 ⊢ ((∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ⊆ ∪ ∪ ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
62	43, 58, 61	syl2anc 584	. . . . . . . 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 138	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ¬ 𝐶 ∈ 𝐾)
64	63	ex 412	. . . . . 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	biimtrid 242	. . 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 1117	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ∪ ciun 4941  ran crn 5620   Fn wfn 6477  ⟶wf 6478  –onto→wfo 6480  ‘cfv 6482  Fincfn 8872  MetOpencmopn 21251

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-en 8873  df-dom 8874  df-fin 8876

This theorem is referenced by:  heiborlem3  37813  heiborlem10  37820