Theorem knatar 6607

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 4173	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
3	2	3ad2ant1 1082	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝐴 ∈ 𝒫 𝐴)
4		simp2 1062	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝐴) ⊆ 𝐴)
5		fveq2 6191	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
6		id 22	. . . . . 6 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
7	5, 6	sseq12d 3634	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝐴))
8	7	intminss 4503	. . . 4 ⊢ ((𝐴 ∈ 𝒫 𝐴 ∧ (𝐹‘𝐴) ⊆ 𝐴) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝐴)
9	3, 4, 8	syl2anc 693	. . 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	syl5eqss 3649	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ 𝐴)
11		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
12		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
13	11, 12	sseq12d 3634	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑤))
14	13	intminss 4503	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝑤)
15	14	adantl 482	. . . . . . . . . . 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)
16	1, 15	syl5eqss 3649	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤)) → 𝑋 ⊆ 𝑤)
17		vex 3203	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18	17	elpw2 4828	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝒫 𝑤 ↔ 𝑋 ⊆ 𝑤)
19	16, 18	sylibr 224	. . . . . . . . 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)
20		simprl 794	. . . . . . . . . 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)
21		simpl3 1066	. . . . . . . . . 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))
22		pweq 4161	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝒫 𝑥 = 𝒫 𝑤)
23		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
24	23	sseq2d 3633	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
25	22, 24	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
26	25	rspcv 3305	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) → ∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
27	20, 21, 26	sylc 65	. . . . . . . . 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))
28		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
29	28	sseq1d 3632	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑤) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑤)))
30	29	rspcv 3305	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝒫 𝑤 → (∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤) → (𝐹‘𝑋) ⊆ (𝐹‘𝑤)))
31	19, 27, 30	sylc 65	. . . . . . . 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))
32		simprr 796	. . . . . . . 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)
33	31, 32	sstrd 3613	. . . . . . 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)
34	33	expr 643	. . . . . 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))
35	34	ralrimiva 2966	. . . . 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))
36		ssintrab 4500	. . . . 5 ⊢ ((𝐹‘𝑋) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ↔ ∀𝑤 ∈ 𝒫 𝐴((𝐹‘𝑤) ⊆ 𝑤 → (𝐹‘𝑋) ⊆ 𝑤))
37	35, 36	sylibr 224	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤})
38	13	cbvrabv 3199	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
39	38	inteqi 4479	. . . . 5 ⊢ ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
40	1, 39	eqtri 2644	. . . 4 ⊢ 𝑋 = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
41	37, 40	syl6sseqr 3652	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝑋)
42	3, 10	sselpwd 4807	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ∈ 𝒫 𝐴)
43		simp3 1063	. . . . . . . . 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))
44		pweq 4161	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
45		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
46	45	sseq2d 3633	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
47	44, 46	raleqbidv 3152	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
48	47	rspcv 3305	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) → ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
49	3, 43, 48	sylc 65	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴))
50	28	sseq1d 3632	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝐴) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝐴)))
51	50	rspcv 3305	. . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴) → (𝐹‘𝑋) ⊆ (𝐹‘𝐴)))
52	42, 49, 51	sylc 65	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ (𝐹‘𝐴))
53	52, 4	sstrd 3613	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝐴)
54		fvex 6201	. . . . . . 7 ⊢ (𝐹‘𝑋) ∈ V
55	54	elpw 4164	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝐴 ↔ (𝐹‘𝑋) ⊆ 𝐴)
56	53, 55	sylibr 224	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ∈ 𝒫 𝐴)
57	54	elpw 4164	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝑋 ↔ (𝐹‘𝑋) ⊆ 𝑋)
58	41, 57	sylibr 224	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ∈ 𝒫 𝑋)
59		pweq 4161	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
60		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
61	60	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
62	59, 61	raleqbidv 3152	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
63	62	rspcv 3305	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) → ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
64	42, 43, 63	sylc 65	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋))
65		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑋) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑋)))
66	65	sseq1d 3632	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑋) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑋) ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
67	66	rspcv 3305	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝑋 → (∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋) → (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
68	58, 64, 67	sylc 65	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋))
69		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → (𝐹‘𝑤) = (𝐹‘(𝐹‘𝑋)))
70		id 22	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → 𝑤 = (𝐹‘𝑋))
71	69, 70	sseq12d 3634	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑋) → ((𝐹‘𝑤) ⊆ 𝑤 ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
72	71	intminss 4503	. . . . 5 ⊢ (((𝐹‘𝑋) ∈ 𝒫 𝐴 ∧ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)) → ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ⊆ (𝐹‘𝑋))
73	56, 68, 72	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ⊆ (𝐹‘𝑋))
74	40, 73	syl5eqss 3649	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ (𝐹‘𝑋))
75	41, 74	eqssd 3620	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) = 𝑋)
76	10, 75	jca 554	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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ∩ cint 4475  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)