gneispace - Mathbox for Richard Penner

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Theorem gneispace 41744

Description: The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.)

**Hypothesis**
Ref	Expression
gneispace.a	⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}

**Assertion**
Ref	Expression
gneispace	⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))))

Distinct variable groups: 𝑛,𝐹,𝑝,𝑓 𝐹,𝑠,𝑓 𝑓,𝑛,𝑝 𝑉,𝑝 Allowed substitution hints: 𝐴(𝑓,𝑛,𝑠,𝑝) 𝑉(𝑓,𝑛,𝑠)

Proof of Theorem gneispace Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gneispace.a	. . 3 ⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}
2	1	gneispace3 41743	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
3		simpll 764	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → Fun 𝐹)
4		simplr 766	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
5		difss 4066	. . . . . 6 ⊢ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅})
6		difss 4066	. . . . . . 7 ⊢ (𝒫 dom 𝐹 ∖ {∅}) ⊆ 𝒫 dom 𝐹
7	6	sspwi 4547	. . . . . 6 ⊢ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) ⊆ 𝒫 𝒫 dom 𝐹
8	5, 7	sstri 3930	. . . . 5 ⊢ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ⊆ 𝒫 𝒫 dom 𝐹
9	4, 8	sstrdi 3933	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
10		simpr 485	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
11		simpl 483	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → Fun 𝐹)
12		fvelrn 6954	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ∈ ran 𝐹)
13	11, 12	sylan 580	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ∈ ran 𝐹)
14		ssel2 3916	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
15		eldifsni 4723	. . . . . . . 8 ⊢ ((𝐹‘𝑝) ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → (𝐹‘𝑝) ≠ ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ≠ ∅)
17	10, 13, 16	syl2an2r 682	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ≠ ∅)
18	17	ralrimiva 3103	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
19		r19.26 3095	. . . . . 6 ⊢ (∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) ↔ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
20	19	biimpri 227	. . . . 5 ⊢ ((∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
21	18, 20	sylan 580	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
22	3, 9, 21	3jca 1127	. . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
23		simp1 1135	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → Fun 𝐹)
24		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑝Fun 𝐹
25		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑝ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹
26		nfra1 3144	. . . . . . . . . 10 ⊢ Ⅎ𝑝∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
27	24, 25, 26	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑝(Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
28		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
29		simpl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) → 𝑝 ∈ 𝑛)
30	29	19.8ad 2175	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) → ∃𝑝 𝑝 ∈ 𝑛)
31	30	ralimi 3087	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) → ∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛)
32	28, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛)
33	32	ralimi 3087	. . . . . . . . . . . . . 14 ⊢ (∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛)
34	33	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛)
35		rsp 3131	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (𝑝 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛))
36	34, 35	syl 17	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛))
37		df-ex 1783	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛)
38	37	ralbii 3092	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ∀𝑛 ∈ (𝐹‘𝑝) ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛)
39		ralnex 3167	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝐹‘𝑝) ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛 ↔ ¬ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
40	38, 39	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
41		0el 4294	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ (𝐹‘𝑝) ↔ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
42	40, 41	xchbinxr 335	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∅ ∈ (𝐹‘𝑝))
43	42	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → ¬ ∅ ∈ (𝐹‘𝑝))
44		elinel1 4129	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) → ∅ ∈ (𝐹‘𝑝))
45	43, 44	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → ¬ ∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
46		disjsn 4647	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
47	45, 46	sylibr 233	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅)
48		disjdif2 4413	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅ → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
49	47, 48	syl 17	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
50	36, 49	syl6 35	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹)))
51		simp2 1136	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
52	12	ex 413	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑝 ∈ dom 𝐹 → (𝐹‘𝑝) ∈ ran 𝐹))
53	23, 52	syl 17	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → (𝐹‘𝑝) ∈ ran 𝐹))
54		ssel2 3916	. . . . . . . . . . . . 13 ⊢ ((ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹)
55		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑝) ∈ V
56	55	elpw 4537	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹 ↔ (𝐹‘𝑝) ⊆ 𝒫 dom 𝐹)
57		df-ss 3904	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑝) ⊆ 𝒫 dom 𝐹 ↔ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
58	56, 57	sylbb 218	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹 → ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
59	54, 58	syl 17	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ (𝐹‘𝑝) ∈ ran 𝐹) → ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
60	51, 53, 59	syl6an 681	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝)))
61	50, 60	jcad 513	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → ((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∧ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))))
62		eqtr 2761	. . . . . . . . . . 11 ⊢ (((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∧ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝)) → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = (𝐹‘𝑝))
63		df-ss 3904	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ ((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (𝐹‘𝑝))
64		indif2 4204	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅})
65	64	eqeq1i 2743	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (𝐹‘𝑝) ↔ (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = (𝐹‘𝑝))
66	63, 65	bitri 274	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = (𝐹‘𝑝))
67	62, 66	sylibr 233	. . . . . . . . . 10 ⊢ (((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∧ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝)) → (𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}))
68	61, 67	syl6 35	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (𝑝 ∈ dom 𝐹 → (𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
69	27, 68	ralrimi 3141	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}))
70	23	funfnd 6465	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → 𝐹 Fn dom 𝐹)
71		sseq1 3946	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑝) → (𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ (𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
72	71	ralrn 6964	. . . . . . . . 9 ⊢ (𝐹 Fn dom 𝐹 → (∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
73	70, 72	syl 17	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
74	69, 73	mpbird 256	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}))
75		pwssb 5030	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}))
76	74, 75	sylibr 233	. . . . . 6 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}))
77		simpl 483	. . . . . . . . . 10 ⊢ (((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → (𝐹‘𝑝) ≠ ∅)
78	77	ralimi 3087	. . . . . . . . 9 ⊢ (∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
79	78	3ad2ant3 1134	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
80	23, 79	jca 512	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (Fun 𝐹 ∧ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅))
81		elrnrexdm 6965	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (∅ ∈ ran 𝐹 → ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝)))
82		nesym 3000	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑝) ≠ ∅ ↔ ¬ ∅ = (𝐹‘𝑝))
83	82	ralbii 3092	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑝 ∈ dom 𝐹 ¬ ∅ = (𝐹‘𝑝))
84		ralnex 3167	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ dom 𝐹 ¬ ∅ = (𝐹‘𝑝) ↔ ¬ ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝))
85	83, 84	sylbb 218	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ → ¬ ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝))
86	81, 85	nsyli 157	. . . . . . . . 9 ⊢ (Fun 𝐹 → (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ → ¬ ∅ ∈ ran 𝐹))
87	86	imp 407	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅) → ¬ ∅ ∈ ran 𝐹)
88		disjsn 4647	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝐹)
89	87, 88	sylibr 233	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅) → (ran 𝐹 ∩ {∅}) = ∅)
90	80, 89	syl 17	. . . . . 6 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (ran 𝐹 ∩ {∅}) = ∅)
91		reldisj 4385	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) → ((ran 𝐹 ∩ {∅}) = ∅ ↔ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
92	91	biimpd 228	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) → ((ran 𝐹 ∩ {∅}) = ∅ → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
93	76, 90, 92	sylc 65	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
94	23, 93	jca 512	. . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
95	19	biimpi 215	. . . . . 6 ⊢ (∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
96	95	3ad2ant3 1134	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
97		simpr 485	. . . . 5 ⊢ ((∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
98	96, 97	syl 17	. . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
99	94, 98	jca 512	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))) → ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
100	22, 99	impbii 208	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
101	2, 100	bitrdi 287	1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 dom cdm 5589 ran crn 5590 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441

This theorem is referenced by: gneispacef2 41746 gneispacern2 41749 gneispace0nelrn 41750

W3C validator