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Theorem ismrc 36741
Description: A function is a Moore closure operator iff it satisfies mrcssid 16198, mrcss 16197, and mrcidm 16200. (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 16188 . . . . 5 mrCls Fn ran Moore
2 fnfun 5946 . . . . 5 (mrCls Fn ran Moore → Fun mrCls)
31, 2ax-mp 5 . . . 4 Fun mrCls
4 fvelima 6205 . . . 4 ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
53, 4mpan 705 . . 3 (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6 elfvex 6178 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7 eqid 2621 . . . . . . . 8 (mrCls‘𝑧) = (mrCls‘𝑧)
87mrcf 16190 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵𝑧)
9 mresspw 16173 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
108, 9fssd 6014 . . . . . 6 (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
117mrcssid 16198 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
1211adantrr 752 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
137mrcss 16197 . . . . . . . . . . 11 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦𝑥𝑥𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14133expb 1263 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦𝑥𝑥𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
1514ancom2s 843 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
167mrcidm 16200 . . . . . . . . . 10 ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1716adantrr 752 . . . . . . . . 9 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
1812, 15, 173jca 1240 . . . . . . . 8 ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥𝐵𝑦𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
1918ex 450 . . . . . . 7 (𝑧 ∈ (Moore‘𝐵) → ((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
2019alrimivv 1853 . . . . . 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 1240 . . . . 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 5983 . . . . . 6 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵𝐹:𝒫 𝐵⟶𝒫 𝐵))
23 fveq1 6147 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹𝑥))
2423sseq2d 3612 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹𝑥)))
25 fveq1 6147 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹𝑦))
2625, 23sseq12d 3613 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑥)))
27 id 22 . . . . . . . . . . 11 ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
2827, 23fveq12d 6154 . . . . . . . . . 10 ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹𝑥)))
2928, 23eqeq12d 2636 . . . . . . . . 9 ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))
3024, 26, 293anbi123d 1396 . . . . . . . 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 330 . . . . . . 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 1848 . . . . . 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 1399 . . . . 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 3020 . . 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 1059 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐵 ∈ V)
38 simp2 1060 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39 ssid 3603 . . . . . . 7 𝑧𝑧
40 3simpb 1057 . . . . . . . . . . 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 1737 . . . . . . . . 9 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))))
43 vex 3189 . . . . . . . . . 10 𝑧 ∈ V
44 sseq1 3605 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4544adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥𝐵𝑧𝐵))
46 sseq12 3607 . . . . . . . . . . . . . 14 ((𝑦 = 𝑧𝑥 = 𝑧) → (𝑦𝑥𝑧𝑧))
4746ancoms 469 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑦𝑥𝑧𝑧))
4845, 47anbi12d 746 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑧𝑧)))
49 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
50 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5149, 50sseq12d 3613 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5251adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5350fveq2d 6152 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5453, 50eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5554adantr 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5652, 55anbi12d 746 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5748, 56imbi12d 334 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑧) → (((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) ↔ ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5857spc2gv 3282 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ((𝑧𝐵𝑧𝑧) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))))
5943, 43, 58mp2an 707 . . . . . . . . 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 1082 . . . . . . 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 712 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → (𝑧𝐵 → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
6362imp 445 . . . . 5 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → (𝑧 ⊆ (𝐹𝑧) ∧ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6463simpld 475 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
65 simp2 1060 . . . . . . . . 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 1737 . . . . . . 7 (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
68673ad2ant3 1082 . . . . . 6 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
69 vex 3189 . . . . . . 7 𝑤 ∈ V
7044adantr 481 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
71 sseq12 3607 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑥 = 𝑧) → (𝑦𝑥𝑤𝑧))
7271ancoms 469 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝑥𝑤𝑧))
7370, 72anbi12d 746 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝑥) ↔ (𝑧𝐵𝑤𝑧)))
74 fveq2 6148 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
75 sseq12 3607 . . . . . . . . . 10 (((𝐹𝑦) = (𝐹𝑤) ∧ (𝐹𝑥) = (𝐹𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7674, 50, 75syl2anr 495 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑤) ⊆ (𝐹𝑧)))
7773, 76imbi12d 334 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) ↔ ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7877spc2gv 3282 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))))
7943, 69, 78mp2an 707 . . . . . 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 1259 . . . 4 (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) ∧ 𝑧𝐵𝑤𝑧) → (𝐹𝑤) ⊆ (𝐹𝑧))
8263simprd 479 . . . 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 36739 . . 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 36738 . . . 4 ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
85 fvssunirn 6174 . . . . . 6 (Moore‘𝐵) ⊆ ran Moore
86 fndm 5948 . . . . . . 7 (mrCls Fn ran Moore → dom mrCls = ran Moore)
871, 86ax-mp 5 . . . . . 6 dom mrCls = ran Moore
8885, 87sseqtr4i 3617 . . . . 5 (Moore‘𝐵) ⊆ dom mrCls
89 funfvima2 6447 . . . . 5 ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
903, 88, 89mp2an 707 . . . 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 2698 . 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 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   I cid 4984  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  Moorecmre 16163  mrClscmrc 16164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168
This theorem is referenced by: (None)
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