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Theorem ismrc 37581
Description: A function is a Moore closure operator iff it satisfies mrcssid 16324, mrcss 16323, and mrcidm 16326. (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 16314 . . . . 5 mrCls Fn ran Moore
2 fnfun 6026 . . . . 5 (mrCls Fn ran Moore → Fun mrCls)
31, 2ax-mp 5 . . . 4 Fun mrCls
4 fvelima 6287 . . . 4 ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
53, 4mpan 706 . . 3 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6 elfvex 6259 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7 eqid 2651 . . . . . . . 8 (mrCls‘𝑧) = (mrCls‘𝑧)
87mrcf 16316 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵𝑧)
9 mresspw 16299 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
108, 9fssd 6095 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
117mrcssid 16324 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
1211adantrr 753 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
137mrcss 16323 . . . . . . . . . . 11 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦𝑥𝑥𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14133expb 1285 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦𝑥𝑥𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
1514ancom2s 861 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
167mrcidm 16326 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1716adantrr 753 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1812, 15, 173jca 1261 . . . . . . . 8 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
1918ex 449 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
2019alrimivv 1896 . . . . . 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 1261 . . . . 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 6064 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵𝐹:𝒫 𝐵⟶𝒫 𝐵))
23 fveq1 6228 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹𝑥))
2423sseq2d 3666 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹𝑥)))
25 fveq1 6228 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹𝑦))
2625, 23sseq12d 3667 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑥)))
27 id 22 . . . . . . . . . . 11 ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
2827, 23fveq12d 6235 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹𝑥)))
2928, 23eqeq12d 2666 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
3024, 26, 293anbi123d 1439 . . . . . . . 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 329 . . . . . . 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 1891 . . . . . 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 1442 . . . . 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 235 . . . 4 (𝑧 ∈ (Moore‘𝐵) → ((mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))))
3534rexlimiv 3056 . . 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 1081 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐵 ∈ V)
38 simp2 1082 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39 ssid 3657 . . . . . . 7 𝑧𝑧
40 3simpb 1079 . . . . . . . . . . 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 1780 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
43 vex 3234 . . . . . . . . . 10 𝑧 ∈ V
44 sseq1 3659 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4544adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥𝐵𝑧𝐵))
46 sseq12 3661 . . . . . . . . . . . . . 14 ((𝑦 = 𝑧𝑥 = 𝑧) → (𝑦𝑥𝑧𝑧))
4746ancoms 468 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑦𝑥𝑧𝑧))
4845, 47anbi12d 747 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑧𝑧)))
49 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
50 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5149, 50sseq12d 3667 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5251adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5350fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5453, 50eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5554adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5652, 55anbi12d 747 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5748, 56imbi12d 333 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑧) → (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) ↔ ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5857spc2gv 3327 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5943, 43, 58mp2an 708 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6042, 59syl 17 . . . . . . . 8 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
61603ad2ant3 1104 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6239, 61mpan2i 713 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (𝑧𝐵 → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6362imp 444 . . . . 5 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6463simpld 474 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
65 simp2 1082 . . . . . . . . 9 ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) → (𝐹𝑦) ⊆ (𝐹𝑥))
6665imim2i 16 . . . . . . . 8 (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
67662alimi 1780 . . . . . . 7 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
68673ad2ant3 1104 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
69 vex 3234 . . . . . . 7 𝑤 ∈ V
7044adantr 480 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
71 sseq12 3661 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑥 = 𝑧) → (𝑦𝑥𝑤𝑧))
7271ancoms 468 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝑥𝑤𝑧))
7370, 72anbi12d 747 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑤𝑧)))
74 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
75 sseq12 3661 . . . . . . . . . 10 (((𝐹𝑦) = (𝐹𝑤) ∧ (𝐹𝑥) = (𝐹𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7674, 50, 75syl2anr 494 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7773, 76imbi12d 333 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) ↔ ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7877spc2gv 3327 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7943, 69, 78mp2an 708 . . . . . 6 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧)))
8068, 79syl 17 . . . . 5 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧)))
81803impib 1281 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))
8263simprd 478 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
8337, 38, 64, 81, 82ismrcd2 37579 . . 3 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
8437, 38, 64, 81, 82ismrcd1 37578 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
85 fvssunirn 6255 . . . . . 6 (Moore‘𝐵) ⊆ ran Moore
86 fndm 6028 . . . . . . 7 (mrCls Fn ran Moore → dom mrCls = ran Moore)
871, 86ax-mp 5 . . . . . 6 dom mrCls = ran Moore
8885, 87sseqtr4i 3671 . . . . 5 (Moore‘𝐵) ⊆ dom mrCls
89 funfvima2 6533 . . . . 5 ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
903, 88, 89mp2an 708 . . . 4 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
9184, 90syl 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)))
9283, 91eqeltrd 2730 . 2 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹 ∈ (mrCls “ (Moore‘𝐵)))
9336, 92impbii 199 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 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   I cid 5052  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  Moorecmre 16289  mrClscmrc 16290
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-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
This theorem is referenced by: (None)
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