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Theorem ismrc 41010
Description: A function is a Moore closure operator iff it satisfies mrcssid 17497, mrcss 17496, and mrcidm 17499. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
ismrc (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ismrc
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnmrc 17487 . . . . 5 mrCls Fn ran Moore
2 fnfun 6602 . . . . 5 (mrCls Fn ran Moore → Fun mrCls)
31, 2ax-mp 5 . . . 4 Fun mrCls
4 fvelima 6908 . . . 4 ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
53, 4mpan 688 . . 3 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6 elfvex 6880 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7 eqid 2736 . . . . . . . 8 (mrCls‘𝑧) = (mrCls‘𝑧)
87mrcf 17489 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵𝑧)
9 mresspw 17472 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
108, 9fssd 6686 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
117mrcssid 17497 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
1211adantrr 715 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
137mrcss 17496 . . . . . . . . . . 11 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦𝑥𝑥𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14133expb 1120 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦𝑥𝑥𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
1514ancom2s 648 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
167mrcidm 17499 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1716adantrr 715 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1812, 15, 173jca 1128 . . . . . . . 8 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
1918ex 413 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
2019alrimivv 1931 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
216, 10, 203jca 1128 . . . . 5 (𝑧 ∈ (Moore‘𝐵) → (𝐵 ∈ V ∧ (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))))
22 feq1 6649 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵𝐹:𝒫 𝐵⟶𝒫 𝐵))
23 fveq1 6841 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹𝑥))
2423sseq2d 3976 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹𝑥)))
25 fveq1 6841 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹𝑦))
2625, 23sseq12d 3977 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑥)))
27 id 22 . . . . . . . . . . 11 ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
2827, 23fveq12d 6849 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹𝑥)))
2928, 23eqeq12d 2752 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
3024, 26, 293anbi123d 1436 . . . . . . . 8 ((mrCls‘𝑧) = 𝐹 → ((𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)) ↔ (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
3130imbi2d 340 . . . . . . 7 ((mrCls‘𝑧) = 𝐹 → (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))) ↔ ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
32312albidv 1926 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
3322, 323anbi23d 1439 . . . . 5 ((mrCls‘𝑧) = 𝐹 → ((𝐵 ∈ V ∧ (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))))
3421, 33syl5ibcom 244 . . . 4 (𝑧 ∈ (Moore‘𝐵) → ((mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))))
3534rexlimiv 3145 . . 3 (∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
365, 35syl 17 . 2 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
37 simp1 1136 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐵 ∈ V)
38 simp2 1137 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39 ssid 3966 . . . . . . 7 𝑧𝑧
40 3simpb 1149 . . . . . . . . . . 11 ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
4140imim2i 16 . . . . . . . . . 10 (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
42412alimi 1814 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
43 sseq1 3969 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4443adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥𝐵𝑧𝐵))
45 sseq12 3971 . . . . . . . . . . . . . 14 ((𝑦 = 𝑧𝑥 = 𝑧) → (𝑦𝑥𝑧𝑧))
4645ancoms 459 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑦𝑥𝑧𝑧))
4744, 46anbi12d 631 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑧𝑧)))
48 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
49 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5048, 49sseq12d 3977 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5150adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
52 2fveq3 6847 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5352, 49eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5453adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5551, 54anbi12d 631 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5647, 55imbi12d 344 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑧) → (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) ↔ ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5756spc2gv 3559 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5857el2v 3453 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5942, 58syl 17 . . . . . . . 8 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
60593ad2ant3 1135 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6139, 60mpan2i 695 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (𝑧𝐵 → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6261imp 407 . . . . 5 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6362simpld 495 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
64 simp2 1137 . . . . . . . . 9 ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) → (𝐹𝑦) ⊆ (𝐹𝑥))
6564imim2i 16 . . . . . . . 8 (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
66652alimi 1814 . . . . . . 7 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
67663ad2ant3 1135 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
6843adantr 481 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
69 sseq12 3971 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑥 = 𝑧) → (𝑦𝑥𝑤𝑧))
7069ancoms 459 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝑥𝑤𝑧))
7168, 70anbi12d 631 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑤𝑧)))
72 fveq2 6842 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
73 sseq12 3971 . . . . . . . . . 10 (((𝐹𝑦) = (𝐹𝑤) ∧ (𝐹𝑥) = (𝐹𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7472, 49, 73syl2anr 597 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7571, 74imbi12d 344 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) ↔ ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7675spc2gv 3559 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7776el2v 3453 . . . . . 6 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧)))
7867, 77syl 17 . . . . 5 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧)))
79783impib 1116 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))
8062simprd 496 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
8137, 38, 63, 79, 80ismrcd2 41008 . . 3 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
8237, 38, 63, 79, 80ismrcd1 41007 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
83 fvssunirn 6875 . . . . . 6 (Moore‘𝐵) ⊆ ran Moore
841fndmi 6606 . . . . . 6 dom mrCls = ran Moore
8583, 84sseqtrri 3981 . . . . 5 (Moore‘𝐵) ⊆ dom mrCls
86 funfvima2 7181 . . . . 5 ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
873, 85, 86mp2an 690 . . . 4 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
8882, 87syl 17 . . 3 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
8981, 88eqeltrd 2838 . 2 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 ∈ (mrCls “ (Moore‘𝐵)))
9036, 89impbii 208 1 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865   I cid 5530  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  Moorecmre 17462  mrClscmrc 17463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-mre 17466  df-mrc 17467
This theorem is referenced by: (None)
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