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Theorem ismrc 39813
 Description: A function is a Moore closure operator iff it satisfies mrcssid 16900, mrcss 16899, and mrcidm 16902. (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 16890 . . . . 5 mrCls Fn ran Moore
2 fnfun 6431 . . . . 5 (mrCls Fn ran Moore → Fun mrCls)
31, 2ax-mp 5 . . . 4 Fun mrCls
4 fvelima 6716 . . . 4 ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
53, 4mpan 689 . . 3 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6 elfvex 6688 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7 eqid 2798 . . . . . . . 8 (mrCls‘𝑧) = (mrCls‘𝑧)
87mrcf 16892 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵𝑧)
9 mresspw 16875 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
108, 9fssd 6510 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
117mrcssid 16900 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
1211adantrr 716 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
137mrcss 16899 . . . . . . . . . . 11 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦𝑥𝑥𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14133expb 1117 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦𝑥𝑥𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
1514ancom2s 649 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
167mrcidm 16902 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1716adantrr 716 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1812, 15, 173jca 1125 . . . . . . . 8 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
1918ex 416 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
2019alrimivv 1929 . . . . . 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 1125 . . . . 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 6476 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵𝐹:𝒫 𝐵⟶𝒫 𝐵))
23 fveq1 6654 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹𝑥))
2423sseq2d 3949 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹𝑥)))
25 fveq1 6654 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹𝑦))
2625, 23sseq12d 3950 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑥)))
27 id 22 . . . . . . . . . . 11 ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
2827, 23fveq12d 6662 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹𝑥)))
2928, 23eqeq12d 2814 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
3024, 26, 293anbi123d 1433 . . . . . . . 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 344 . . . . . . 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 1924 . . . . . 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 1436 . . . . 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 248 . . . 4 (𝑧 ∈ (Moore‘𝐵) → ((mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))))
3534rexlimiv 3240 . . 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 1133 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐵 ∈ V)
38 simp2 1134 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39 ssid 3939 . . . . . . 7 𝑧𝑧
40 3simpb 1146 . . . . . . . . . . 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 3942 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4443adantr 484 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥𝐵𝑧𝐵))
45 sseq12 3944 . . . . . . . . . . . . . 14 ((𝑦 = 𝑧𝑥 = 𝑧) → (𝑦𝑥𝑧𝑧))
4645ancoms 462 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑦𝑥𝑧𝑧))
4744, 46anbi12d 633 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑧𝑧)))
48 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
49 fveq2 6655 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5048, 49sseq12d 3950 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5150adantr 484 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
52 2fveq3 6660 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5352, 49eqeq12d 2814 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5453adantr 484 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5551, 54anbi12d 633 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5647, 55imbi12d 348 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑧) → (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) ↔ ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5756spc2gv 3550 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5857el2v 3449 . . . . . . . . 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 1132 . . . . . . 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 696 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (𝑧𝐵 → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6261imp 410 . . . . 5 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6362simpld 498 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
64 simp2 1134 . . . . . . . . 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 1132 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
6843adantr 484 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
69 sseq12 3944 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑥 = 𝑧) → (𝑦𝑥𝑤𝑧))
7069ancoms 462 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝑥𝑤𝑧))
7168, 70anbi12d 633 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑤𝑧)))
72 fveq2 6655 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
73 sseq12 3944 . . . . . . . . . 10 (((𝐹𝑦) = (𝐹𝑤) ∧ (𝐹𝑥) = (𝐹𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7472, 49, 73syl2anr 599 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7571, 74imbi12d 348 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) ↔ ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7675spc2gv 3550 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7776el2v 3449 . . . . . 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 1113 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))
8062simprd 499 . . . 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 39811 . . 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 39810 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
83 fvssunirn 6684 . . . . . 6 (Moore‘𝐵) ⊆ ran Moore
841fndmi 6434 . . . . . 6 dom mrCls = ran Moore
8583, 84sseqtrri 3954 . . . . 5 (Moore‘𝐵) ⊆ dom mrCls
86 funfvima2 6981 . . . . 5 ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
873, 85, 86mp2an 691 . . . 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 2890 . 2 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 ∈ (mrCls “ (Moore‘𝐵)))
9036, 89impbii 212 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 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ∪ cuni 4804   I cid 5428  dom cdm 5523  ran crn 5524   “ cima 5526  Fun wfun 6326   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  Moorecmre 16865  mrClscmrc 16866 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-int 4843  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-mre 16869  df-mrc 16870 This theorem is referenced by: (None)
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