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Theorem ismrc 42712
Description: A function is a Moore closure operator iff it satisfies mrcssid 17660, mrcss 17659, and mrcidm 17662. (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 17650 . . . . 5 mrCls Fn ran Moore
2 fnfun 6668 . . . . 5 (mrCls Fn ran Moore → Fun mrCls)
31, 2ax-mp 5 . . . 4 Fun mrCls
4 fvelima 6974 . . . 4 ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
53, 4mpan 690 . . 3 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6 elfvex 6944 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7 eqid 2737 . . . . . . . 8 (mrCls‘𝑧) = (mrCls‘𝑧)
87mrcf 17652 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵𝑧)
9 mresspw 17635 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
108, 9fssd 6753 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
117mrcssid 17660 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
1211adantrr 717 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
137mrcss 17659 . . . . . . . . . . 11 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦𝑥𝑥𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14133expb 1121 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦𝑥𝑥𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
1514ancom2s 650 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
167mrcidm 17662 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1716adantrr 717 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1812, 15, 173jca 1129 . . . . . . . 8 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
1918ex 412 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
2019alrimivv 1928 . . . . . 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 1129 . . . . 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 6716 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵𝐹:𝒫 𝐵⟶𝒫 𝐵))
23 fveq1 6905 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹𝑥))
2423sseq2d 4016 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹𝑥)))
25 fveq1 6905 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹𝑦))
2625, 23sseq12d 4017 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑥)))
27 id 22 . . . . . . . . . . 11 ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
2827, 23fveq12d 6913 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹𝑥)))
2928, 23eqeq12d 2753 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
3024, 26, 293anbi123d 1438 . . . . . . . 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 1923 . . . . . 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 1441 . . . . 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 245 . . . 4 (𝑧 ∈ (Moore‘𝐵) → ((mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))))
3534rexlimiv 3148 . . 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 1137 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐵 ∈ V)
38 simp2 1138 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39 ssid 4006 . . . . . . 7 𝑧𝑧
40 3simpb 1150 . . . . . . . . . . 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 1812 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
43 sseq1 4009 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4443adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥𝐵𝑧𝐵))
45 sseq12 4011 . . . . . . . . . . . . . 14 ((𝑦 = 𝑧𝑥 = 𝑧) → (𝑦𝑥𝑧𝑧))
4645ancoms 458 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑦𝑥𝑧𝑧))
4744, 46anbi12d 632 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑧𝑧)))
48 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
49 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5048, 49sseq12d 4017 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5150adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
52 2fveq3 6911 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5352, 49eqeq12d 2753 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5453adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5551, 54anbi12d 632 . . . . . . . . . . . 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 3600 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5857el2v 3487 . . . . . . . . 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 1136 . . . . . . 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 697 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (𝑧𝐵 → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6261imp 406 . . . . 5 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6362simpld 494 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
64 simp2 1138 . . . . . . . . 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 1812 . . . . . . 7 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
67663ad2ant3 1136 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
6843adantr 480 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
69 sseq12 4011 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑥 = 𝑧) → (𝑦𝑥𝑤𝑧))
7069ancoms 458 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝑥𝑤𝑧))
7168, 70anbi12d 632 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑤𝑧)))
72 fveq2 6906 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
73 sseq12 4011 . . . . . . . . . 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 3600 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7776el2v 3487 . . . . . 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 1117 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))
8062simprd 495 . . . 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 42710 . . 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 42709 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
83 fvssunirn 6939 . . . . . 6 (Moore‘𝐵) ⊆ ran Moore
841fndmi 6672 . . . . . 6 dom mrCls = ran Moore
8583, 84sseqtrri 4033 . . . . 5 (Moore‘𝐵) ⊆ dom mrCls
86 funfvima2 7251 . . . . 5 ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
873, 85, 86mp2an 692 . . . 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 2841 . 2 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 ∈ (mrCls “ (Moore‘𝐵)))
9036, 89impbii 209 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 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   I cid 5577  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  Moorecmre 17625  mrClscmrc 17626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630
This theorem is referenced by: (None)
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