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Theorem isacs2 16361
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 16359 . 2 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
2 iunss 4593 . . . . . . . . 9 ( 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)
3 ffun 6086 . . . . . . . . . . 11 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → Fun 𝑓)
4 funiunfv 6546 . . . . . . . . . . 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 3665 . . . . . . . . 9 (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ( 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
72, 6syl5rbbr 275 . . . . . . . 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 3015 . . . . . 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 670 . . . . 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 1814 . . . 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 805 . . . . . . . . . . . . 13 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
13 inss1 3866 . . . . . . . . . . . . . . . 16 (𝒫 𝑠 ∩ Fin) ⊆ 𝒫 𝑠
1413sseli 3632 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ 𝒫 𝑠)
15 elpwi 4201 . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . 13 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠𝐶)
19 isacs2.f . . . . . . . . . . . . . 14 𝐹 = (mrCls‘𝐶)
2019mrcsscl 16327 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦𝑠𝑠𝐶) → (𝐹𝑦) ⊆ 𝑠)
2112, 17, 18, 20syl3anc 1366 . . . . . . . . . . . 12 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹𝑦) ⊆ 𝑠)
2221ralrimiva 2995 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)
2322adantlr 751 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑠𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)
2423adantllr 755 . . . . . . . . 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 813 . . . . . . . . . . . . . . . . . 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 4201 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
2827ad2antlr 763 . . . . . . . . . . . . . . . . . . 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 3646 . . . . . . . . . . . . . . . . . 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 16332 . . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
3231elpw 4197 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ 𝒫 (𝐹𝑦) ↔ 𝑦 ⊆ (𝐹𝑦))
3330, 32sylibr 224 . . . . . . . . . . . . . . . 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 3867 . . . . . . . . . . . . . . . . . 18 (𝒫 𝑠 ∩ Fin) ⊆ Fin
3534sseli 3632 . . . . . . . . . . . . . . . . 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 3831 . . . . . . . . . . . . . . 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 16318 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ 𝐶)
3925, 29, 38syl2anc 694 . . . . . . . . . . . . . . . 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 16299 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
4140ad3antrrr 766 . . . . . . . . . . . . . . . . . 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 3637 . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐹𝑦) → (𝑡𝐶 ↔ (𝐹𝑦) ∈ 𝐶))
46 pweq 4194 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (𝐹𝑦) → 𝒫 𝑡 = 𝒫 (𝐹𝑦))
4746ineq1d 3846 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐹𝑦) → (𝒫 𝑡 ∩ Fin) = (𝒫 (𝐹𝑦) ∩ Fin))
48 sseq2 3660 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐹𝑦) → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝑓𝑧) ⊆ (𝐹𝑦)))
4947, 48raleqbidv 3182 . . . . . . . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑓𝑧) = (𝑓𝑦))
5554sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑓𝑧) ⊆ (𝐹𝑦) ↔ (𝑓𝑦) ⊆ (𝐹𝑦)))
5655rspcva 3338 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (𝒫 (𝐹𝑦) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐹𝑦) ∩ Fin)(𝑓𝑧) ⊆ (𝐹𝑦)) → (𝑓𝑦) ⊆ (𝐹𝑦))
5737, 53, 56syl2anc 694 . . . . . . . . . . . . . 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 3643 . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
6362sseq1d 3665 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑓𝑦) ⊆ 𝑠 ↔ (𝑓𝑧) ⊆ 𝑠))
6463cbvralv 3201 . . . . . . . . . . 11 (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑦) ⊆ 𝑠 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠)
6561, 64sylib 208 . . . . . . . . . 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 807 . . . . . . . . . . 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 762 . . . . . . . . . . 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 2718 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → (𝑡𝐶𝑠𝐶))
69 pweq 4194 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠 → 𝒫 𝑡 = 𝒫 𝑠)
7069ineq1d 3846 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝒫 𝑡 ∩ Fin) = (𝒫 𝑠 ∩ Fin))
71 sseq2 3660 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝑓𝑧) ⊆ 𝑠))
7270, 71raleqbidv 3182 . . . . . . . . . . . . 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 3338 . . . . . . . . . . 11 ((𝑠 ∈ 𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡)) → (𝑠𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓𝑧) ⊆ 𝑠))
7566, 67, 74syl2anc 694 . . . . . . . . . 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 247 . . . . . . . . 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 895 . . . . . . . 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 2995 . . . . . . 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 1901 . . . . 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 16316 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
8281, 40fssd 6095 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
83 fvex 6239 . . . . . . . . 9 (mrCls‘𝐶) ∈ V
8419, 83eqeltri 2726 . . . . . . . 8 𝐹 ∈ V
85 feq1 6064 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋𝐹:𝒫 𝑋⟶𝒫 𝑋))
86 fveq1 6228 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
8786sseq1d 3665 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → ((𝑓𝑧) ⊆ 𝑡 ↔ (𝐹𝑧) ⊆ 𝑡))
8887ralbidv 3015 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑧) ⊆ 𝑡))
89 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
9089sseq1d 3665 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝐹𝑧) ⊆ 𝑡 ↔ (𝐹𝑦) ⊆ 𝑡))
9190cbvralv 3201 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡)
9288, 91syl6bb 276 . . . . . . . . . . . 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 3015 . . . . . . . . . 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 3660 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → ((𝐹𝑦) ⊆ 𝑡 ↔ (𝐹𝑦) ⊆ 𝑠))
9670, 95raleqbidv 3182 . . . . . . . . . . . 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 3201 . . . . . . . . . 10 (∀𝑡 ∈ 𝒫 𝑋(𝑡𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹𝑦) ⊆ 𝑡) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
9994, 98syl6bb 276 . . . . . . . . 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 747 . . . . . . . 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 3331 . . . . . . 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 202 . . . 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 272 . . 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 670 . 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 264 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 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   ciun 4552  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  Fincfn 7997  Moorecmre 16289  mrClscmrc 16290  ACScacs 16292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mre 16293  df-mrc 16294  df-acs 16296
This theorem is referenced by:  acsfiel  16362  isacs5  17219
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