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Theorem dssmapnvod 40231
Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is its own inverse, an involution. (Contributed by RP, 20-Apr-2021.)
Hypotheses
Ref Expression
dssmapfvd.o 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
dssmapfvd.d 𝐷 = (𝑂𝐵)
dssmapfvd.b (𝜑𝐵𝑉)
Assertion
Ref Expression
dssmapnvod (𝜑𝐷 = 𝐷)
Distinct variable groups:   𝐵,𝑏,𝑓,𝑠   𝜑,𝑏,𝑓,𝑠
Allowed substitution hints:   𝐷(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapnvod
Dummy variables 𝑔 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . . . 9 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
2 difeq2 4097 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝐵𝑠) = (𝐵𝑡))
32fveq2d 6671 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑓‘(𝐵𝑠)) = (𝑓‘(𝐵𝑡)))
43difeq2d 4103 . . . . . . . . . 10 (𝑠 = 𝑡 → (𝐵 ∖ (𝑓‘(𝐵𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑡))))
54cbvmptv 5166 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) = (𝑡 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑡))))
61, 5syl6eq 2877 . . . . . . . 8 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → 𝑔 = (𝑡 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑡)))))
7 ssun1 4152 . . . . . . . . . . . 12 𝐵 ⊆ (𝐵 ∪ (𝑓‘(𝐵𝑡)))
8 sspwb 5338 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐵 ∪ (𝑓‘(𝐵𝑡))) ↔ 𝒫 𝐵 ⊆ 𝒫 (𝐵 ∪ (𝑓‘(𝐵𝑡))))
97, 8mpbi 231 . . . . . . . . . . 11 𝒫 𝐵 ⊆ 𝒫 (𝐵 ∪ (𝑓‘(𝐵𝑡)))
10 dssmapfvd.b . . . . . . . . . . . 12 (𝜑𝐵𝑉)
11 pwidg 4559 . . . . . . . . . . . 12 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
1210, 11syl 17 . . . . . . . . . . 11 (𝜑𝐵 ∈ 𝒫 𝐵)
139, 12sseldi 3969 . . . . . . . . . 10 (𝜑𝐵 ∈ 𝒫 (𝐵 ∪ (𝑓‘(𝐵𝑡))))
14 fvex 6680 . . . . . . . . . . 11 (𝑓‘(𝐵𝑡)) ∈ V
1514elpwun 7480 . . . . . . . . . 10 (𝐵 ∈ 𝒫 (𝐵 ∪ (𝑓‘(𝐵𝑡))) ↔ (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ 𝒫 𝐵)
1613, 15sylib 219 . . . . . . . . 9 (𝜑 → (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ 𝒫 𝐵)
1716ad2antrr 722 . . . . . . . 8 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ 𝒫 𝐵)
186, 17fmpt3d 6876 . . . . . . 7 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → 𝑔:𝒫 𝐵⟶𝒫 𝐵)
1910pwexd 5277 . . . . . . . . 9 (𝜑 → 𝒫 𝐵 ∈ V)
2019adantr 481 . . . . . . . 8 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → 𝒫 𝐵 ∈ V)
2120, 20elmapd 8410 . . . . . . 7 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ↔ 𝑔:𝒫 𝐵⟶𝒫 𝐵))
2218, 21mpbird 258 . . . . . 6 ((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → 𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵))
2322adantrl 712 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → 𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵))
24 simplr 765 . . . . . . . . . . . 12 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
25 difeq2 4097 . . . . . . . . . . . . . . 15 (𝑠 = 𝑢 → (𝐵𝑠) = (𝐵𝑢))
2625fveq2d 6671 . . . . . . . . . . . . . 14 (𝑠 = 𝑢 → (𝑓‘(𝐵𝑠)) = (𝑓‘(𝐵𝑢)))
2726difeq2d 4103 . . . . . . . . . . . . 13 (𝑠 = 𝑢 → (𝐵 ∖ (𝑓‘(𝐵𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑢))))
2827cbvmptv 5166 . . . . . . . . . . . 12 (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) = (𝑢 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑢))))
2924, 28syl6eq 2877 . . . . . . . . . . 11 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑔 = (𝑢 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑢)))))
30 difeq2 4097 . . . . . . . . . . . . . 14 (𝑢 = (𝐵𝑡) → (𝐵𝑢) = (𝐵 ∖ (𝐵𝑡)))
3130fveq2d 6671 . . . . . . . . . . . . 13 (𝑢 = (𝐵𝑡) → (𝑓‘(𝐵𝑢)) = (𝑓‘(𝐵 ∖ (𝐵𝑡))))
3231difeq2d 4103 . . . . . . . . . . . 12 (𝑢 = (𝐵𝑡) → (𝐵 ∖ (𝑓‘(𝐵𝑢))) = (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))))
3332adantl 482 . . . . . . . . . . 11 ((((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑢 = (𝐵𝑡)) → (𝐵 ∖ (𝑓‘(𝐵𝑢))) = (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))))
34 ssun1 4152 . . . . . . . . . . . . . . 15 𝐵 ⊆ (𝐵𝑡)
35 sspwb 5338 . . . . . . . . . . . . . . 15 (𝐵 ⊆ (𝐵𝑡) ↔ 𝒫 𝐵 ⊆ 𝒫 (𝐵𝑡))
3634, 35mpbi 231 . . . . . . . . . . . . . 14 𝒫 𝐵 ⊆ 𝒫 (𝐵𝑡)
3736, 12sseldi 3969 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ 𝒫 (𝐵𝑡))
38 vex 3503 . . . . . . . . . . . . . 14 𝑡 ∈ V
3938elpwun 7480 . . . . . . . . . . . . 13 (𝐵 ∈ 𝒫 (𝐵𝑡) ↔ (𝐵𝑡) ∈ 𝒫 𝐵)
4037, 39sylib 219 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
4140ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
42 difexg 5228 . . . . . . . . . . . . 13 (𝐵𝑉 → (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
4310, 42syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
4443ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
4529, 33, 41, 44fvmptd 6771 . . . . . . . . . 10 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑔‘(𝐵𝑡)) = (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))))
4645difeq2d 4103 . . . . . . . . 9 (((𝜑𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵𝑡))) = (𝐵 ∖ (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡))))))
4746adantlrl 716 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵𝑡))) = (𝐵 ∖ (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡))))))
48 elpwi 4554 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
49 dfss4 4239 . . . . . . . . . . . . 13 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
5048, 49sylib 219 . . . . . . . . . . . 12 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
5150fveq2d 6671 . . . . . . . . . . 11 (𝑡 ∈ 𝒫 𝐵 → (𝑓‘(𝐵 ∖ (𝐵𝑡))) = (𝑓𝑡))
5251difeq2d 4103 . . . . . . . . . 10 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡)))) = (𝐵 ∖ (𝑓𝑡)))
5352difeq2d 4103 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡))))) = (𝐵 ∖ (𝐵 ∖ (𝑓𝑡))))
5453adantl 482 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑓‘(𝐵 ∖ (𝐵𝑡))))) = (𝐵 ∖ (𝐵 ∖ (𝑓𝑡))))
5519, 19elmapd 8410 . . . . . . . . . . . . 13 (𝜑 → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↔ 𝑓:𝒫 𝐵⟶𝒫 𝐵))
5655biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝑓:𝒫 𝐵⟶𝒫 𝐵)
5756ffvelrnda 6847 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑓𝑡) ∈ 𝒫 𝐵)
5857elpwid 4556 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑓𝑡) ⊆ 𝐵)
59 dfss4 4239 . . . . . . . . . 10 ((𝑓𝑡) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝑓𝑡))) = (𝑓𝑡))
6058, 59sylib 219 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑓𝑡))) = (𝑓𝑡))
6160adantlrr 717 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑓𝑡))) = (𝑓𝑡))
6247, 54, 613eqtrrd 2866 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑓𝑡) = (𝐵 ∖ (𝑔‘(𝐵𝑡))))
6362ralrimiva 3187 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → ∀𝑡 ∈ 𝒫 𝐵(𝑓𝑡) = (𝐵 ∖ (𝑔‘(𝐵𝑡))))
64 elmapfn 8419 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝑓 Fn 𝒫 𝐵)
6564ad2antrl 724 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → 𝑓 Fn 𝒫 𝐵)
66 difeq2 4097 . . . . . . . . 9 (𝑡 = 𝑧 → (𝐵𝑡) = (𝐵𝑧))
6766fveq2d 6671 . . . . . . . 8 (𝑡 = 𝑧 → (𝑔‘(𝐵𝑡)) = (𝑔‘(𝐵𝑧)))
6867difeq2d 4103 . . . . . . 7 (𝑡 = 𝑧 → (𝐵 ∖ (𝑔‘(𝐵𝑡))) = (𝐵 ∖ (𝑔‘(𝐵𝑧))))
69 difexg 5228 . . . . . . . . 9 (𝐵𝑉 → (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ V)
7010, 69syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ V)
7170ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ V)
72 difexg 5228 . . . . . . . . 9 (𝐵𝑉 → (𝐵 ∖ (𝑔‘(𝐵𝑧))) ∈ V)
7310, 72syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∖ (𝑔‘(𝐵𝑧))) ∈ V)
7473ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵𝑧))) ∈ V)
7565, 68, 71, 74fnmptfvd 6807 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → (𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑓𝑡) = (𝐵 ∖ (𝑔‘(𝐵𝑡)))))
7663, 75mpbird 258 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))
7723, 76jca 512 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))) → (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))))
78 simpr 485 . . . . . . . . 9 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))
79 difeq2 4097 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (𝐵𝑧) = (𝐵𝑡))
8079fveq2d 6671 . . . . . . . . . . 11 (𝑧 = 𝑡 → (𝑔‘(𝐵𝑧)) = (𝑔‘(𝐵𝑡)))
8180difeq2d 4103 . . . . . . . . . 10 (𝑧 = 𝑡 → (𝐵 ∖ (𝑔‘(𝐵𝑧))) = (𝐵 ∖ (𝑔‘(𝐵𝑡))))
8281cbvmptv 5166 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))) = (𝑡 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑡))))
8378, 82syl6eq 2877 . . . . . . . 8 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → 𝑓 = (𝑡 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑡)))))
84 ssun1 4152 . . . . . . . . . . . 12 𝐵 ⊆ (𝐵 ∪ (𝑔‘(𝐵𝑡)))
85 sspwb 5338 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐵 ∪ (𝑔‘(𝐵𝑡))) ↔ 𝒫 𝐵 ⊆ 𝒫 (𝐵 ∪ (𝑔‘(𝐵𝑡))))
8684, 85mpbi 231 . . . . . . . . . . 11 𝒫 𝐵 ⊆ 𝒫 (𝐵 ∪ (𝑔‘(𝐵𝑡)))
8786, 12sseldi 3969 . . . . . . . . . 10 (𝜑𝐵 ∈ 𝒫 (𝐵 ∪ (𝑔‘(𝐵𝑡))))
88 fvex 6680 . . . . . . . . . . 11 (𝑔‘(𝐵𝑡)) ∈ V
8988elpwun 7480 . . . . . . . . . 10 (𝐵 ∈ 𝒫 (𝐵 ∪ (𝑔‘(𝐵𝑡))) ↔ (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ 𝒫 𝐵)
9087, 89sylib 219 . . . . . . . . 9 (𝜑 → (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ 𝒫 𝐵)
9190ad2antrr 722 . . . . . . . 8 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵𝑡))) ∈ 𝒫 𝐵)
9283, 91fmpt3d 6876 . . . . . . 7 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → 𝑓:𝒫 𝐵⟶𝒫 𝐵)
9319adantr 481 . . . . . . . 8 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → 𝒫 𝐵 ∈ V)
9493, 93elmapd 8410 . . . . . . 7 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↔ 𝑓:𝒫 𝐵⟶𝒫 𝐵))
9592, 94mpbird 258 . . . . . 6 ((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) → 𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵))
9695adantrl 712 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → 𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵))
97 simplr 765 . . . . . . . . . . . 12 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))
98 difeq2 4097 . . . . . . . . . . . . . . 15 (𝑧 = 𝑢 → (𝐵𝑧) = (𝐵𝑢))
9998fveq2d 6671 . . . . . . . . . . . . . 14 (𝑧 = 𝑢 → (𝑔‘(𝐵𝑧)) = (𝑔‘(𝐵𝑢)))
10099difeq2d 4103 . . . . . . . . . . . . 13 (𝑧 = 𝑢 → (𝐵 ∖ (𝑔‘(𝐵𝑧))) = (𝐵 ∖ (𝑔‘(𝐵𝑢))))
101100cbvmptv 5166 . . . . . . . . . . . 12 (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))) = (𝑢 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑢))))
10297, 101syl6eq 2877 . . . . . . . . . . 11 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑓 = (𝑢 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑢)))))
10330fveq2d 6671 . . . . . . . . . . . . 13 (𝑢 = (𝐵𝑡) → (𝑔‘(𝐵𝑢)) = (𝑔‘(𝐵 ∖ (𝐵𝑡))))
104103difeq2d 4103 . . . . . . . . . . . 12 (𝑢 = (𝐵𝑡) → (𝐵 ∖ (𝑔‘(𝐵𝑢))) = (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))))
105104adantl 482 . . . . . . . . . . 11 ((((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑢 = (𝐵𝑡)) → (𝐵 ∖ (𝑔‘(𝐵𝑢))) = (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))))
10640ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
107 difexg 5228 . . . . . . . . . . . . 13 (𝐵𝑉 → (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
10810, 107syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
109108ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))) ∈ V)
110102, 105, 106, 109fvmptd 6771 . . . . . . . . . 10 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑓‘(𝐵𝑡)) = (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))))
111110difeq2d 4103 . . . . . . . . 9 (((𝜑𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵𝑡))) = (𝐵 ∖ (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡))))))
112111adantlrl 716 . . . . . . . 8 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵𝑡))) = (𝐵 ∖ (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡))))))
11350fveq2d 6671 . . . . . . . . . . 11 (𝑡 ∈ 𝒫 𝐵 → (𝑔‘(𝐵 ∖ (𝐵𝑡))) = (𝑔𝑡))
114113difeq2d 4103 . . . . . . . . . 10 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡)))) = (𝐵 ∖ (𝑔𝑡)))
115114difeq2d 4103 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡))))) = (𝐵 ∖ (𝐵 ∖ (𝑔𝑡))))
116115adantl 482 . . . . . . . 8 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑔‘(𝐵 ∖ (𝐵𝑡))))) = (𝐵 ∖ (𝐵 ∖ (𝑔𝑡))))
11719, 19elmapd 8410 . . . . . . . . . . . . 13 (𝜑 → (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ↔ 𝑔:𝒫 𝐵⟶𝒫 𝐵))
118117biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝑔:𝒫 𝐵⟶𝒫 𝐵)
119118ffvelrnda 6847 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑔𝑡) ∈ 𝒫 𝐵)
120119elpwid 4556 . . . . . . . . . 10 (((𝜑𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑔𝑡) ⊆ 𝐵)
121 dfss4 4239 . . . . . . . . . 10 ((𝑔𝑡) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝑔𝑡))) = (𝑔𝑡))
122120, 121sylib 219 . . . . . . . . 9 (((𝜑𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑔𝑡))) = (𝑔𝑡))
123122adantlrr 717 . . . . . . . 8 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝑔𝑡))) = (𝑔𝑡))
124112, 116, 1233eqtrrd 2866 . . . . . . 7 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑔𝑡) = (𝐵 ∖ (𝑓‘(𝐵𝑡))))
125124ralrimiva 3187 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → ∀𝑡 ∈ 𝒫 𝐵(𝑔𝑡) = (𝐵 ∖ (𝑓‘(𝐵𝑡))))
126 elmapfn 8419 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝑔 Fn 𝒫 𝐵)
127126ad2antrl 724 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → 𝑔 Fn 𝒫 𝐵)
128 difeq2 4097 . . . . . . . . 9 (𝑡 = 𝑠 → (𝐵𝑡) = (𝐵𝑠))
129128fveq2d 6671 . . . . . . . 8 (𝑡 = 𝑠 → (𝑓‘(𝐵𝑡)) = (𝑓‘(𝐵𝑠)))
130129difeq2d 4103 . . . . . . 7 (𝑡 = 𝑠 → (𝐵 ∖ (𝑓‘(𝐵𝑡))) = (𝐵 ∖ (𝑓‘(𝐵𝑠))))
131 difexg 5228 . . . . . . . . 9 (𝐵𝑉 → (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ V)
13210, 131syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ V)
133132ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵𝑡))) ∈ V)
134 difexg 5228 . . . . . . . . 9 (𝐵𝑉 → (𝐵 ∖ (𝑓‘(𝐵𝑠))) ∈ V)
13510, 134syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∖ (𝑓‘(𝐵𝑠))) ∈ V)
136135ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝑓‘(𝐵𝑠))) ∈ V)
137127, 130, 133, 136fnmptfvd 6807 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → (𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑔𝑡) = (𝐵 ∖ (𝑓‘(𝐵𝑡)))))
138125, 137mpbird 258 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
13996, 138jca 512 . . . 4 ((𝜑 ∧ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))) → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
14077, 139impbida 797 . . 3 (𝜑 → ((𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑔 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ↔ (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ∧ 𝑓 = (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧)))))))
141140mptcnv 5996 . 2 (𝜑(𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) = (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))))
142 dssmapfvd.o . . . 4 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
143 dssmapfvd.d . . . 4 𝐷 = (𝑂𝐵)
144142, 143, 10dssmapfvd 40228 . . 3 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
145144cnveqd 5745 . 2 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
146 fveq1 6666 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓‘(𝑏𝑠)) = (𝑔‘(𝑏𝑠)))
147146difeq2d 4103 . . . . . . . 8 (𝑓 = 𝑔 → (𝑏 ∖ (𝑓‘(𝑏𝑠))) = (𝑏 ∖ (𝑔‘(𝑏𝑠))))
148147mpteq2dv 5159 . . . . . . 7 (𝑓 = 𝑔 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑠)))))
149 difeq2 4097 . . . . . . . . . 10 (𝑠 = 𝑧 → (𝑏𝑠) = (𝑏𝑧))
150149fveq2d 6671 . . . . . . . . 9 (𝑠 = 𝑧 → (𝑔‘(𝑏𝑠)) = (𝑔‘(𝑏𝑧)))
151150difeq2d 4103 . . . . . . . 8 (𝑠 = 𝑧 → (𝑏 ∖ (𝑔‘(𝑏𝑠))) = (𝑏 ∖ (𝑔‘(𝑏𝑧))))
152151cbvmptv 5166 . . . . . . 7 (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑠)))) = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑧))))
153148, 152syl6eq 2877 . . . . . 6 (𝑓 = 𝑔 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑧)))))
154153cbvmptv 5166 . . . . 5 (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑔 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑧)))))
155154mpteq2i 5155 . . . 4 (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))))) = (𝑏 ∈ V ↦ (𝑔 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑧))))))
156142, 155eqtri 2849 . . 3 𝑂 = (𝑏 ∈ V ↦ (𝑔 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑔‘(𝑏𝑧))))))
157156, 143, 10dssmapfvd 40228 . 2 (𝜑𝐷 = (𝑔 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑧 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑔‘(𝐵𝑧))))))
158141, 145, 1573eqtr4d 2871 1 (𝜑𝐷 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  cdif 3937  cun 3938  wss 3940  𝒫 cpw 4542  cmpt 5143  ccnv 5553   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by:  dssmapf1od  40232  dssmap2d  40233  ntrclsnvobr  40267  clsneicnv  40320  neicvgnvo  40330  dssmapclsntr  40344
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