gneispace - Mathbox for Richard Penner

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Theorem gneispace 44717

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 44716	. 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 778	. . . 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 780	. . . . 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 4092	. . . . . 6 ⊢ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅})
6		difss 4092	. . . . . . 7 ⊢ (𝒫 dom 𝐹 ∖ {∅}) ⊆ 𝒫 dom 𝐹
7	6	sspwi 4570	. . . . . 6 ⊢ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) ⊆ 𝒫 𝒫 dom 𝐹
8	5, 7	sstri 3948	. . . . 5 ⊢ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ⊆ 𝒫 𝒫 dom 𝐹
9	4, 8	sstrdi 3951	. . . 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 489	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
11		simpl 487	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → Fun 𝐹)
12		fvelrn 7061	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ∈ ran 𝐹)
13	11, 12	sylan 591	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ∈ ran 𝐹)
14		ssel2 3934	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
15		eldifsni 4753	. . . . . . . 8 ⊢ ((𝐹‘𝑝) ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → (𝐹‘𝑝) ≠ ∅)
16	14, 15	syl 18	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ≠ ∅)
17	10, 13, 16	syl2an2r 697	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ 𝑝 ∈ dom 𝐹) → (𝐹‘𝑝) ≠ ∅)
18	17	ralrimiva 3157	. . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
19		r19.26 3125	. . . . . 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 231	. . . . 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 591	. . . 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 1144	. . 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 1152	. . . . 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 1937	. . . . . . . . . 10 ⊢ Ⅎ𝑝Fun 𝐹
25		nfv 1937	. . . . . . . . . 10 ⊢ Ⅎ𝑝ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹
26		nfra1 3289	. . . . . . . . . 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 1924	. . . . . . . . 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 489	. . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) → 𝑝 ∈ 𝑛)
30	29	19.8ad 2220	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) → ∃𝑝 𝑝 ∈ 𝑛)
31	30	ralimi 3102	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 3102	. . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . 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 3253	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (𝑝 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛))
36	34, 35	syl 18	. . . . . . . . . . . 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 1803	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛)
38	37	ralbii 3111	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ∀𝑛 ∈ (𝐹‘𝑝) ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛)
39		ralnex 3091	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝐹‘𝑝) ¬ ∀𝑝 ¬ 𝑝 ∈ 𝑛 ↔ ¬ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
40	38, 39	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
41		0el 4319	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ (𝐹‘𝑝) ↔ ∃𝑛 ∈ (𝐹‘𝑝)∀𝑝 ¬ 𝑝 ∈ 𝑛)
42	40, 41	xchbinxr 338	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 ↔ ¬ ∅ ∈ (𝐹‘𝑝))
43	42	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → ¬ ∅ ∈ (𝐹‘𝑝))
44		elinel1 4156	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) → ∅ ∈ (𝐹‘𝑝))
45	43, 44	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → ¬ ∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
46		disjsn 4673	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
47	45, 46	sylibr 237	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅)
48		disjdif2 4437	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∩ {∅}) = ∅ → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
49	47, 48	syl 18	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (𝐹‘𝑝)∃𝑝 𝑝 ∈ 𝑛 → (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹))
50	36, 49	syl6 36	. . . . . . . . . . 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 1153	. . . . . . . . . . . 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 417	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑝 ∈ dom 𝐹 → (𝐹‘𝑝) ∈ ran 𝐹))
53	23, 52	syl 18	. . . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ ((ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ (𝐹‘𝑝) ∈ ran 𝐹) → (𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹)
55		fvex 6884	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑝) ∈ V
56	55	elpw 4562	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹 ↔ (𝐹‘𝑝) ⊆ 𝒫 dom 𝐹)
57		dfss2 3925	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑝) ⊆ 𝒫 dom 𝐹 ↔ ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
58	56, 57	sylbb 222	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑝) ∈ 𝒫 𝒫 dom 𝐹 → ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
59	54, 58	syl 18	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ (𝐹‘𝑝) ∈ ran 𝐹) → ((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) = (𝐹‘𝑝))
60	51, 53, 59	syl6an 696	. . . . . . . . . . 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 521	. . . . . . . . . 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 2785	. . . . . . . . . . 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		dfss2 3925	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ ((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (𝐹‘𝑝))
64		indif2 4236	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅})
65	64	eqeq1i 2770	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑝) ∩ (𝒫 dom 𝐹 ∖ {∅})) = (𝐹‘𝑝) ↔ (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = (𝐹‘𝑝))
66	63, 65	bitri 278	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ (((𝐹‘𝑝) ∩ 𝒫 dom 𝐹) ∖ {∅}) = (𝐹‘𝑝))
67	62, 66	sylibr 237	. . . . . . . . . 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 36	. . . . . . . . 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 3263	. . . . . . . 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 6556	. . . . . . . . 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 3964	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑝) → (𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ (𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
72	71	ralrn 7073	. . . . . . . . 9 ⊢ (𝐹 Fn dom 𝐹 → (∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}) ↔ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ⊆ (𝒫 dom 𝐹 ∖ {∅})))
73	70, 72	syl 18	. . . . . . . 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 260	. . . . . . 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 5062	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ⊆ (𝒫 dom 𝐹 ∖ {∅}))
76	74, 75	sylibr 237	. . . . . 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 487	. . . . . . . . . 10 ⊢ (((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))) → (𝐹‘𝑝) ≠ ∅)
78	77	ralimi 3102	. . . . . . . . 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 1151	. . . . . . . 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 520	. . . . . . 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 7074	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (∅ ∈ ran 𝐹 → ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝)))
82		nesym 3016	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑝) ≠ ∅ ↔ ¬ ∅ = (𝐹‘𝑝))
83	82	ralbii 3111	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑝 ∈ dom 𝐹 ¬ ∅ = (𝐹‘𝑝))
84		ralnex 3091	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ dom 𝐹 ¬ ∅ = (𝐹‘𝑝) ↔ ¬ ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝))
85	83, 84	sylbb 222	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ → ¬ ∃𝑝 ∈ dom 𝐹∅ = (𝐹‘𝑝))
86	81, 85	nsyli 158	. . . . . . . . 9 ⊢ (Fun 𝐹 → (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ → ¬ ∅ ∈ ran 𝐹))
87	86	imp 411	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅) → ¬ ∅ ∈ ran 𝐹)
88		disjsn 4673	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝐹)
89	87, 88	sylibr 237	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅) → (ran 𝐹 ∩ {∅}) = ∅)
90	80, 89	syl 18	. . . . . 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 4410	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) → ((ran 𝐹 ∩ {∅}) = ∅ ↔ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
92	91	biimpd 232	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝒫 (𝒫 dom 𝐹 ∖ {∅}) → ((ran 𝐹 ∩ {∅}) = ∅ → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
93	76, 90, 92	sylc 66	. . . . 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 520	. . . 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 219	. . . . . 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 1151	. . . . 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 489	. . . . 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 18	. . . 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 520	. . 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 212	. 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 290	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 209 ∧ wa 400 ∧ w3a 1101 ∀wal 1561 = wceq 1563 ∃wex 1802 ∈ wcel 2145 {cab 2743 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 ∅c0 4288 𝒫 cpw 4558 {csn 4585 dom cdm 5651 ran crn 5652 Fun wfun 6519 Fn wfn 6520 ⟶wf 6521 ‘cfv 6525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pr 5394

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-fv 6533

This theorem is referenced by: gneispacef2 44719 gneispacern2 44722 gneispace0nelrn 44723

W3C validator