Theorem knatar 7377

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 4581	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝐴 ∈ 𝒫 𝐴)
4		simp2 1155	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝐴) ⊆ 𝐴)
5		fveq2 6896	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
6		id 23	. . . . . 6 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
7	5, 6	sseq12d 3946	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝐴))
8	7	intminss 4937	. . . 4 ⊢ ((𝐴 ∈ 𝒫 𝐴 ∧ (𝐹‘𝐴) ⊆ 𝐴) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝐴)
9	3, 4, 8	syl2anc 596	. . 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 3951	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ 𝐴)
11		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
12	11	sseq1d 3944	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑤) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑤)))
13		pweq 4574	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → 𝒫 𝑥 = 𝒫 𝑤)
14		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
15	14	sseq2d 3945	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
16	13, 15	raleqbidv 3338	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑤(𝐹‘𝑦) ⊆ (𝐹‘𝑤)))
17		simpl3 1212	. . . . . . . . . 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 783	. . . . . . . . . 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 3581	. . . . . . . . 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 6896	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
21		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
22	20, 21	sseq12d 3946	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑤))
23	22	intminss 4937	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝒫 𝐴 ∧ (𝐹‘𝑤) ⊆ 𝑤) → ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⊆ 𝑤)
24	23	adantl 487	. . . . . . . . . . 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 3951	. . . . . . . . . 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 3458	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
27	26	elpw2 5300	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝒫 𝑤 ↔ 𝑋 ⊆ 𝑤)
28	25, 27	sylibr 237	. . . . . . . . 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 3581	. . . . . . . 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 785	. . . . . . . 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 3923	. . . . . . 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 462	. . . . . 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 3158	. . . . 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 4934	. . . . 5 ⊢ ((𝐹‘𝑋) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ↔ ∀𝑤 ∈ 𝒫 𝐴((𝐹‘𝑤) ⊆ 𝑤 → (𝐹‘𝑋) ⊆ 𝑤))
35	33, 34	sylibr 237	. . . 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 3426	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
37	36	inteqi 4914	. . . . 5 ⊢ ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
38	1, 37	eqtri 2786	. . . 4 ⊢ 𝑋 = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤}
39	35, 38	sseqtrrdi 3954	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝑋)
40	11	sseq1d 3944	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝐴) ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝐴)))
41		pweq 4574	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
42		fveq2 6896	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
43	42	sseq2d 3945	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
44	41, 43	raleqbidv 3338	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝐹‘𝑦) ⊆ (𝐹‘𝐴)))
45		simp3 1156	. . . . . . . . 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 3581	. . . . . . . 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 5294	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ∈ 𝒫 𝐴)
48	40, 46, 47	rspcdva 3581	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ (𝐹‘𝐴))
49	48, 4	sstrd 3923	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ⊆ 𝐴)
50		fvex 6909	. . . . . . 7 ⊢ (𝐹‘𝑋) ∈ V
51	50	elpw 4564	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝐴 ↔ (𝐹‘𝑋) ⊆ 𝐴)
52	49, 51	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) ∈ 𝒫 𝐴)
53		fveq2 6896	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑋) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑋)))
54	53	sseq1d 3944	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑋) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑋) ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
55		pweq 4574	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
56		fveq2 6896	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
57	56	sseq2d 3945	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
58	55, 57	raleqbidv 3338	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑋(𝐹‘𝑦) ⊆ (𝐹‘𝑋)))
59	58, 45, 47	rspcdva 3581	. . . . . 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 4564	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ 𝒫 𝑋 ↔ (𝐹‘𝑋) ⊆ 𝑋)
61	39, 60	sylibr 237	. . . . . 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 3581	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋))
63		fveq2 6896	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → (𝐹‘𝑤) = (𝐹‘(𝐹‘𝑋)))
64		id 23	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑋) → 𝑤 = (𝐹‘𝑋))
65	63, 64	sseq12d 3946	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑋) → ((𝐹‘𝑤) ⊆ 𝑤 ↔ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)))
66	65	intminss 4937	. . . . 5 ⊢ (((𝐹‘𝑋) ∈ 𝒫 𝐴 ∧ (𝐹‘(𝐹‘𝑋)) ⊆ (𝐹‘𝑋)) → ∩ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑤) ⊆ 𝑤} ⊆ (𝐹‘𝑋))
67	52, 62, 66	syl2anc 596	. . . 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 3951	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → 𝑋 ⊆ (𝐹‘𝑋))
69	39, 68	eqssd 3930	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝐹‘𝑋) = 𝑋)
70	10, 69	jca 521	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  {crab 3416   ⊆ wss 3900  𝒫 cpw 4560  ∩ cint 4910  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-iota 6501  df-fv 6555

This theorem is used by: (None)