Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isacs2 Structured version   Visualization version   GIF version

Theorem isacs2 16137
 Description: In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
isacs2.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
isacs2 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
Distinct variable groups:   𝐶,𝑠,𝑦   𝐹,𝑠,𝑦   𝑋,𝑠,𝑦

Proof of Theorem isacs2
Dummy variables 𝑓 𝑡 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isacs 16135 . 2 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
2 iunss 4497 . . . . . . . . 9 ( 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)
3 ffun 5961 . . . . . . . . . . 11 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → Fun 𝑓)
4 funiunfv 6410 . . . . . . . . . . 11 (Fun 𝑓 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) = (𝑓 “ (𝒫 𝑡 ∩ Fin)))
53, 4syl 17 . . . . . . . . . 10 (𝑓:𝒫 𝑋⟶𝒫 𝑋 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) = (𝑓 “ (𝒫 𝑡 ∩ Fin)))
65sseq1d 3595 . . . . . . . . 9 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ( 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
72, 6syl5rbbr 274 . . . . . . . 8 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ( (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))
87bibi2d 331 . . . . . . 7 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ((𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ (𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
98ralbidv 2969 . . . . . 6 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
109pm5.32i 667 . . . . 5 ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
1110exbii 1764 . . . 4 (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
12 simpll 786 . . . . . . . . . . . . 13 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
13 inss1 3795 . . . . . . . . . . . . . . . 16 (𝒫 𝑠 ∩ Fin) ⊆ 𝒫 𝑠
1413sseli 3564 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ 𝒫 𝑠)
15 elpwi 4117 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝑠𝑦𝑠)
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦𝑠)
1716adantl 481 . . . . . . . . . . . . 13 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦𝑠)
18 simplr 788 . . . . . . . . . . . . 13 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠𝐶)
19 isacs2.f . . . . . . . . . . . . . 14 𝐹 = (mrCls‘𝐶)
2019mrcsscl 16103 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦𝑠𝑠𝐶) → (𝐹𝑦) ⊆ 𝑠)
2112, 17, 18, 20syl3anc 1318 . . . . . . . . . . . 12 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹𝑦) ⊆ 𝑠)
2221ralrimiva 2949 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)
2322adantlr 747 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑠𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)
2423adantllr 751 . . . . . . . . 9 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑠𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)
25 simplll 794 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
2616adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦𝑠)
27 elpwi 4117 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
2827ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠𝑋)
2926, 28sstrd 3578 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦𝑋)
3025, 19, 29mrcssidd 16108 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ (𝐹𝑦))
31 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
3231elpw 4114 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ 𝒫 (𝐹𝑦) ↔ 𝑦 ⊆ (𝐹𝑦))
3330, 32sylibr 223 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ 𝒫 (𝐹𝑦))
34 inss2 3796 . . . . . . . . . . . . . . . . . 18 (𝒫 𝑠 ∩ Fin) ⊆ Fin
3534sseli 3564 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ Fin)
3635adantl 481 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ Fin)
3733, 36elind 3760 . . . . . . . . . . . . . . 15 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ (𝒫 (𝐹𝑦) ∩ Fin))
3819mrccl 16094 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ 𝐶)
3925, 29, 38syl2anc 691 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹𝑦) ∈ 𝐶)
40 mresspw 16075 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
4140ad3antrrr 762 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ⊆ 𝒫 𝑋)
4241, 39sseldd 3569 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝑋)
43 simprr 792 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) → ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))
4443ad2antrr 758 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))
45 eleq1 2676 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐹𝑦) → (𝑡𝐶 ↔ (𝐹𝑦) ∈ 𝐶))
46 pweq 4111 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (𝐹𝑦) → 𝒫 𝑡 = 𝒫 (𝐹𝑦))
4746ineq1d 3775 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐹𝑦) → (𝒫 𝑡 ∩ Fin) = (𝒫 (𝐹𝑦) ∩ Fin))
48 sseq2 3590 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐹𝑦) → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝑓𝑧) ⊆ (𝐹𝑦)))
4947, 48raleqbidv 3129 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐹𝑦) → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦)))
5045, 49bibi12d 334 . . . . . . . . . . . . . . . . . 18 (𝑡 = (𝐹𝑦) → ((𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡) ↔ ((𝐹𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦))))
5150rspcva 3280 . . . . . . . . . . . . . . . . 17 (((𝐹𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) → ((𝐹𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦)))
5242, 44, 51syl2anc 691 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ((𝐹𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦)))
5339, 52mpbid 221 . . . . . . . . . . . . . . 15 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦))
54 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑓𝑧) = (𝑓𝑦))
5554sseq1d 3595 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑓𝑧) ⊆ (𝐹𝑦) ↔ (𝑓𝑦) ⊆ (𝐹𝑦)))
5655rspcva 3280 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (𝒫 (𝐹𝑦) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦)) → (𝑓𝑦) ⊆ (𝐹𝑦))
5737, 53, 56syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝑓𝑦) ⊆ (𝐹𝑦))
58 sstr2 3575 . . . . . . . . . . . . . 14 ((𝑓𝑦) ⊆ (𝐹𝑦) → ((𝐹𝑦) ⊆ 𝑠 → (𝑓𝑦) ⊆ 𝑠))
5957, 58syl 17 . . . . . . . . . . . . 13 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ((𝐹𝑦) ⊆ 𝑠 → (𝑓𝑦) ⊆ 𝑠))
6059ralimdva 2945 . . . . . . . . . . . 12 (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠 → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑦) ⊆ 𝑠))
6160imp 444 . . . . . . . . . . 11 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑦) ⊆ 𝑠)
62 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
6362sseq1d 3595 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑓𝑦) ⊆ 𝑠 ↔ (𝑓𝑧) ⊆ 𝑠))
6463cbvralv 3147 . . . . . . . . . . 11 (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑦) ⊆ 𝑠 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠)
6561, 64sylib 207 . . . . . . . . . 10 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠)
66 simplr 788 . . . . . . . . . . 11 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → 𝑠 ∈ 𝒫 𝑋)
6743ad2antrr 758 . . . . . . . . . . 11 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))
68 eleq1 2676 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → (𝑡𝐶𝑠𝐶))
69 pweq 4111 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠 → 𝒫 𝑡 = 𝒫 𝑠)
7069ineq1d 3775 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝒫 𝑡 ∩ Fin) = (𝒫 𝑠 ∩ Fin))
71 sseq2 3590 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝑓𝑧) ⊆ 𝑠))
7270, 71raleqbidv 3129 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠))
7368, 72bibi12d 334 . . . . . . . . . . . 12 (𝑡 = 𝑠 → ((𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡) ↔ (𝑠𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠)))
7473rspcva 3280 . . . . . . . . . . 11 ((𝑠 ∈ 𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) → (𝑠𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠))
7566, 67, 74syl2anc 691 . . . . . . . . . 10 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → (𝑠𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠))
7665, 75mpbird 246 . . . . . . . . 9 ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → 𝑠𝐶)
7724, 76impbida 873 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) → (𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
7877ralrimiva 2949 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
7978ex 449 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
8079exlimdv 1848 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
8119mrcf 16092 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
8281, 40fssd 5970 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
83 fvex 6113 . . . . . . . . 9 (mrCls‘𝐶) ∈ V
8419, 83eqeltri 2684 . . . . . . . 8 𝐹 ∈ V
85 feq1 5939 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋𝐹:𝒫 𝑋⟶𝒫 𝑋))
86 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
8786sseq1d 3595 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝐹𝑧) ⊆ 𝑡))
8887ralbidv 2969 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑧) ⊆ 𝑡))
89 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
9089sseq1d 3595 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) ⊆ 𝑡 ↔ (𝐹𝑦) ⊆ 𝑡))
9190cbvralv 3147 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡)
9288, 91syl6bb 275 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡))
9392bibi2d 331 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡) ↔ (𝑡𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡)))
9493ralbidv 2969 . . . . . . . . . 10 (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡)))
95 sseq2 3590 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → ((𝐹𝑦) ⊆ 𝑡 ↔ (𝐹𝑦) ⊆ 𝑠))
9670, 95raleqbidv 3129 . . . . . . . . . . . 12 (𝑡 = 𝑠 → (∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
9768, 96bibi12d 334 . . . . . . . . . . 11 (𝑡 = 𝑠 → ((𝑡𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡) ↔ (𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
9897cbvralv 3147 . . . . . . . . . 10 (∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
9994, 98syl6bb 275 . . . . . . . . 9 (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
10085, 99anbi12d 743 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) ↔ (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))))
10184, 100spcev 3273 . . . . . . 7 ((𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
10282, 101sylan 487 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)))
103102ex 449 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡))))
10480, 103impbid 201 . . . 4 (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
10511, 104syl5bb 271 . . 3 (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
106105pm5.32i 667 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
1071, 106bitri 263 1 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070  df-acs 16072 This theorem is referenced by:  acsfiel  16138  isacs5  16995
 Copyright terms: Public domain W3C validator