Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  knatar Structured version   Visualization version   GIF version

Theorem knatar 6507
 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 ∣ (𝐹𝑧) ⊆ 𝑧}
Assertion
Ref Expression
knatar ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 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.
StepHypRef Expression
1 knatar.1 . . 3 𝑋 = {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧}
2 pwidg 4121 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
323ad2ant1 1075 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝐴 ∈ 𝒫 𝐴)
4 simp2 1055 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝐴) ⊆ 𝐴)
5 fveq2 6103 . . . . . 6 (𝑧 = 𝐴 → (𝐹𝑧) = (𝐹𝐴))
6 id 22 . . . . . 6 (𝑧 = 𝐴𝑧 = 𝐴)
75, 6sseq12d 3597 . . . . 5 (𝑧 = 𝐴 → ((𝐹𝑧) ⊆ 𝑧 ↔ (𝐹𝐴) ⊆ 𝐴))
87intminss 4438 . . . 4 ((𝐴 ∈ 𝒫 𝐴 ∧ (𝐹𝐴) ⊆ 𝐴) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝐴)
93, 4, 8syl2anc 691 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝐴)
101, 9syl5eqss 3612 . 2 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋𝐴)
11 fveq2 6103 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
12 id 22 . . . . . . . . . . . . . 14 (𝑧 = 𝑤𝑧 = 𝑤)
1311, 12sseq12d 3597 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ((𝐹𝑧) ⊆ 𝑧 ↔ (𝐹𝑤) ⊆ 𝑤))
1413intminss 4438 . . . . . . . . . . . 12 ((𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝑤)
1514adantl 481 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝑤)
161, 15syl5eqss 3612 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑋𝑤)
17 vex 3176 . . . . . . . . . . 11 𝑤 ∈ V
1817elpw2 4755 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑤𝑋𝑤)
1916, 18sylibr 223 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑋 ∈ 𝒫 𝑤)
20 simprl 790 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑤 ∈ 𝒫 𝐴)
21 simpl3 1059 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥))
22 pweq 4111 . . . . . . . . . . . 12 (𝑥 = 𝑤 → 𝒫 𝑥 = 𝒫 𝑤)
23 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
2423sseq2d 3596 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑤)))
2522, 24raleqbidv 3129 . . . . . . . . . . 11 (𝑥 = 𝑤 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤)))
2625rspcv 3278 . . . . . . . . . 10 (𝑤 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤)))
2720, 21, 26sylc 63 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤))
28 fveq2 6103 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2928sseq1d 3595 . . . . . . . . . 10 (𝑦 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝑤) ↔ (𝐹𝑋) ⊆ (𝐹𝑤)))
3029rspcv 3278 . . . . . . . . 9 (𝑋 ∈ 𝒫 𝑤 → (∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤) → (𝐹𝑋) ⊆ (𝐹𝑤)))
3119, 27, 30sylc 63 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑋) ⊆ (𝐹𝑤))
32 simprr 792 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑤) ⊆ 𝑤)
3331, 32sstrd 3578 . . . . . . 7 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑋) ⊆ 𝑤)
3433expr 641 . . . . . 6 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ 𝑤 ∈ 𝒫 𝐴) → ((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
3534ralrimiva 2949 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑤 ∈ 𝒫 𝐴((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
36 ssintrab 4435 . . . . 5 ((𝐹𝑋) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ↔ ∀𝑤 ∈ 𝒫 𝐴((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
3735, 36sylibr 223 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤})
3813cbvrabv 3172 . . . . . 6 {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
3938inteqi 4414 . . . . 5 {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
401, 39eqtri 2632 . . . 4 𝑋 = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
4137, 40syl6sseqr 3615 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ 𝑋)
42 elpw2g 4754 . . . . . . . . . 10 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
43423ad2ant1 1075 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
4410, 43mpbird 246 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋 ∈ 𝒫 𝐴)
45 simp3 1056 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥))
46 pweq 4111 . . . . . . . . . . 11 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
47 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
4847sseq2d 3596 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝐴)))
4946, 48raleqbidv 3129 . . . . . . . . . 10 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴)))
5049rspcv 3278 . . . . . . . . 9 (𝐴 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴)))
513, 45, 50sylc 63 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴))
5228sseq1d 3595 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝐴) ↔ (𝐹𝑋) ⊆ (𝐹𝐴)))
5352rspcv 3278 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴) → (𝐹𝑋) ⊆ (𝐹𝐴)))
5444, 51, 53sylc 63 . . . . . . 7 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ (𝐹𝐴))
5554, 4sstrd 3578 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ 𝐴)
56 fvex 6113 . . . . . . 7 (𝐹𝑋) ∈ V
5756elpw 4114 . . . . . 6 ((𝐹𝑋) ∈ 𝒫 𝐴 ↔ (𝐹𝑋) ⊆ 𝐴)
5855, 57sylibr 223 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ∈ 𝒫 𝐴)
5956elpw 4114 . . . . . . 7 ((𝐹𝑋) ∈ 𝒫 𝑋 ↔ (𝐹𝑋) ⊆ 𝑋)
6041, 59sylibr 223 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ∈ 𝒫 𝑋)
61 pweq 4111 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
62 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
6362sseq2d 3596 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑋)))
6461, 63raleqbidv 3129 . . . . . . . 8 (𝑥 = 𝑋 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋)))
6564rspcv 3278 . . . . . . 7 (𝑋 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋)))
6644, 45, 65sylc 63 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋))
67 fveq2 6103 . . . . . . . 8 (𝑦 = (𝐹𝑋) → (𝐹𝑦) = (𝐹‘(𝐹𝑋)))
6867sseq1d 3595 . . . . . . 7 (𝑦 = (𝐹𝑋) → ((𝐹𝑦) ⊆ (𝐹𝑋) ↔ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
6968rspcv 3278 . . . . . 6 ((𝐹𝑋) ∈ 𝒫 𝑋 → (∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋) → (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
7060, 66, 69sylc 63 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋))
71 fveq2 6103 . . . . . . 7 (𝑤 = (𝐹𝑋) → (𝐹𝑤) = (𝐹‘(𝐹𝑋)))
72 id 22 . . . . . . 7 (𝑤 = (𝐹𝑋) → 𝑤 = (𝐹𝑋))
7371, 72sseq12d 3597 . . . . . 6 (𝑤 = (𝐹𝑋) → ((𝐹𝑤) ⊆ 𝑤 ↔ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
7473intminss 4438 . . . . 5 (((𝐹𝑋) ∈ 𝒫 𝐴 ∧ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)) → {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ⊆ (𝐹𝑋))
7558, 70, 74syl2anc 691 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ⊆ (𝐹𝑋))
7640, 75syl5eqss 3612 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋 ⊆ (𝐹𝑋))
7741, 76eqssd 3585 . 2 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) = 𝑋)
7810, 77jca 553 1 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ∩ cint 4410  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator