Theorem knatar 7260

Description: The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)

Hypothesis

Ref	Expression
knatar.1	⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧}

Assertion

Ref	Expression
knatar	⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐹,𝑦,𝑧   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑋(𝑧)

Proof of Theorem knatar

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		knatar.1	. . 3 ⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧}
2		pwidg 4559	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝐴 ∈ 𝒫 𝐴)
4		simp2 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝐴) ⊆ 𝐴)
5		fveq2 6804	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
6		id 22	. . . . . 6 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
7	5, 6	sseq12d 3959	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝐴))
8	7	intminss 4912	. . . 4 ⊢ ((𝐴 ∈ 𝒫 𝐴 ∧ (𝐹‘𝐴) ⊆ 𝐴) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝐴)
9	3, 4, 8	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝐴)
10	1, 9	eqsstrid 3974	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ 𝐴)
11		fveq2 6804	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
12	11	sseq1d 3957	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑤) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑤)))
13		pweq 4553	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → 𝒫 𝑥 = 𝒫 𝑤)
14		fveq2 6804	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
15	14	sseq2d 3958	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
16	13, 15	raleqbidv 3348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
17		simpl3 1193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥))
18		simprl 769	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → 𝑤 ∈ 𝒫 𝐴)
19	16, 17, 18	rspcdva 3567	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → ∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤))
20		fveq2 6804	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
21		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
22	20, 21	sseq12d 3959	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑤))
23	22	intminss 4912	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝑤)
24	23	adantl 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝑤)
25	1, 24	eqsstrid 3974	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → 𝑋 ⊆ 𝑤)
26		vex 3441	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
27	26	elpw2 5278	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝒫 𝑤 ↔ 𝑋 ⊆ 𝑤)
28	25, 27	sylibr 233	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → 𝑋 ∈ 𝒫 𝑤)
29	12, 19, 28	rspcdva 3567	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → (𝐹‘𝑋) ⊆ (𝐹‘𝑤))
30		simprr 771	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → (𝐹‘𝑤) ⊆ 𝑤)
31	29, 30	sstrd 3936	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → (𝐹‘𝑋) ⊆ 𝑤)
32	31	expr 458	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ 𝑤 ∈ 𝒫 𝐴) → ((𝐹‘𝑤) ⊆ 𝑤 → (𝐹‘𝑋) ⊆ 𝑤))
33	32	ralrimiva 3140	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑤 ∈ 𝒫 𝐴((𝐹‘𝑤) ⊆ 𝑤 → (𝐹‘𝑋) ⊆ 𝑤))
34		ssintrab 4909	. . . . 5 ⊢ ((𝐹‘𝑋) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ↔ ∀𝑤 ∈ 𝒫 𝐴((𝐹‘𝑤) ⊆ 𝑤 → (𝐹‘𝑋) ⊆ 𝑤))
35	33, 34	sylibr 233	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤})
36	22	cbvrabv 3433	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
37	36	inteqi 4890	. . . . 5 ⊢ ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
38	1, 37	eqtri 2764	. . . 4 ⊢ 𝑋 = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
39	35, 38	sseqtrrdi 3977	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝑋)
40	11	sseq1d 3957	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝐴) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝐴)))
41		pweq 4553	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
42		fveq2 6804	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
43	42	sseq2d 3958	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
44	41, 43	raleqbidv 3348	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
45		simp3 1138	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥))
46	44, 45, 3	rspcdva 3567	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴))
47	3, 10	sselpwd 5259	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ∈ 𝒫 𝐴)
48	40, 46, 47	rspcdva 3567	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ (𝐹‘𝐴))
49	48, 4	sstrd 3936	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝐴)
50		fvex 6817	. . . . . . 7 ⊢ (𝐹‘𝑋) ∈ V
51	50	elpw 4543	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝐴 ↔ (𝐹‘𝑋) ⊆ 𝐴)
52	49, 51	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ∈ 𝒫 𝐴)
53		fveq2 6804	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑋) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑋)))
54	53	sseq1d 3957	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑋) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑋) ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
55		pweq 4553	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
56		fveq2 6804	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
57	56	sseq2d 3958	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
58	55, 57	raleqbidv 3348	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
59	58, 45, 47	rspcdva 3567	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋))
60	50	elpw 4543	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝑋 ↔ (𝐹‘𝑋) ⊆ 𝑋)
61	39, 60	sylibr 233	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ∈ 𝒫 𝑋)
62	54, 59, 61	rspcdva 3567	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋))
63		fveq2 6804	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → (𝐹‘𝑤) = (𝐹‘(𝐹‘𝑋)))
64		id 22	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → 𝑤 = (𝐹‘𝑋))
65	63, 64	sseq12d 3959	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑋) → ((𝐹‘𝑤) ⊆ 𝑤 ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
66	65	intminss 4912	. . . . 5 ⊢ (((𝐹‘𝑋) ∈ 𝒫 𝐴 ∧ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)) → ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ⊆ (𝐹‘𝑋))
67	52, 62, 66	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ⊆ (𝐹‘𝑋))
68	38, 67	eqsstrid 3974	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ (𝐹‘𝑋))
69	39, 68	eqssd 3943	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) = 𝑋)
70	10, 69	jca 513	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∀wral 3062  {crab 3284   ⊆ wss 3892  𝒫 cpw 4539  ∩ cint 4886  ‘cfv 6458

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-br 5082  df-iota 6410  df-fv 6466

This theorem is referenced by: (None)