Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrclsk3 Structured version   Visualization version   GIF version

Theorem ntrclsk3 37388
 Description: The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclsk3 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
Distinct variable groups:   𝐵,𝑠,𝑡,𝑖,𝑗,𝑘   𝐼,𝑠,𝑡,𝑖,𝑗,𝑘   𝜑,𝑠,𝑡,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑡,𝑖,𝑗,𝑘,𝑠)   𝐾(𝑡,𝑖,𝑗,𝑘,𝑠)   𝑂(𝑡,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
21ineq1d 3775 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
3 ineq1 3769 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
43fveq2d 6107 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
52, 4sseq12d 3597 . . 3 (𝑠 = 𝑎 → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡))))
6 fveq2 6103 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
76ineq2d 3776 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
8 ineq2 3770 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
98fveq2d 6107 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
107, 9sseq12d 3597 . . 3 (𝑡 = 𝑏 → (((𝐼𝑎) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑎𝑡)) ↔ ((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏))))
115, 10cbvral2v 3155 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)))
12 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
13 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1412, 13ntrclsbex 37352 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 3700 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 4734 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4117 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
19 simpl 472 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → 𝐵 ∈ V)
20 difssd 3700 . . . . . 6 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 4734 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 simpr 476 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → 𝑠 = (𝐵𝑎))
2322difeq2d 3690 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2423eqeq2d 2620 . . . . . 6 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
25 eqcom 2617 . . . . . 6 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2624, 25syl6bb 275 . . . . 5 (((𝐵 ∈ V ∧ 𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
27 dfss4 3820 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 205 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 481 . . . . 5 ((𝐵 ∈ V ∧ 𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3289 . . . 4 ((𝐵 ∈ V ∧ 𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3114, 18, 30syl2an 493 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 simpl1 1057 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
33 difssd 3700 . . . . . 6 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
3414, 33sselpwd 4734 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
3532, 34syl 17 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
36 elpwi 4117 . . . . . 6 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
37 simpl 472 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → 𝐵 ∈ V)
38 difssd 3700 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ⊆ 𝐵)
3937, 38sselpwd 4734 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
40 simpr 476 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → 𝑡 = (𝐵𝑏))
4140difeq2d 3690 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4241eqeq2d 2620 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
43 eqcom 2617 . . . . . . . 8 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4442, 43syl6bb 275 . . . . . . 7 (((𝐵 ∈ V ∧ 𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
45 dfss4 3820 . . . . . . . . 9 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4645biimpi 205 . . . . . . . 8 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4746adantl 481 . . . . . . 7 ((𝐵 ∈ V ∧ 𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
4839, 44, 47rspcedvd 3289 . . . . . 6 ((𝐵 ∈ V ∧ 𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
4914, 36, 48syl2an 493 . . . . 5 ((𝜑𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
50493ad2antl1 1216 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
51 simp13 1086 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑎 = (𝐵𝑠))
52 fveq2 6103 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
5352ineq1d 3775 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
54 ineq1 3769 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
5554fveq2d 6107 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
5653, 55sseq12d 3597 . . . . . 6 (𝑎 = (𝐵𝑠) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏))))
5751, 56syl 17 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏))))
58 fveq2 6103 . . . . . . . 8 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
5958ineq2d 3776 . . . . . . 7 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
60 ineq2 3770 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
61 difundi 3838 . . . . . . . . 9 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6260, 61syl6eqr 2662 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6362fveq2d 6107 . . . . . . 7 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
6459, 63sseq12d 3597 . . . . . 6 (𝑏 = (𝐵𝑡) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
65643ad2ant3 1077 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ⊆ (𝐼‘((𝐵𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
66 simp11 1084 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
67 ntrcls.o . . . . . . . . . 10 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
6867, 12, 13ntrclsiex 37371 . . . . . . . . 9 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6968, 14jca 553 . . . . . . . 8 (𝜑 → (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V))
7066, 69syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V))
71 elmapi 7765 . . . . . . . . . . . 12 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7271adantr 480 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
73 simpr 476 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
74 difssd 3700 . . . . . . . . . . . 12 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
7573, 74sselpwd 4734 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
7672, 75ffvelrnd 6268 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
7776elpwid 4118 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
78 orc 399 . . . . . . . . 9 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵))
79 inss 3804 . . . . . . . . 9 (((𝐼‘(𝐵𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵𝑡)) ⊆ 𝐵) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
8077, 78, 793syl 18 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
81 difssd 3700 . . . . . . . . . . 11 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
8273, 81sselpwd 4734 . . . . . . . . . 10 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8372, 82ffvelrnd 6268 . . . . . . . . 9 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8483elpwid 4118 . . . . . . . 8 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
8580, 84jca 553 . . . . . . 7 ((𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵))
86 sscon34b 37337 . . . . . . 7 ((((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
8770, 85, 863syl 18 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
88 difindi 3840 . . . . . . . 8 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
8988sseq2i 3593 . . . . . . 7 ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
9089a1i 11 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))))
9166, 14syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐵 ∈ V)
9266, 68syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
93 simp12 1085 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
94 rp-simp2 37107 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
95 simpl2 1058 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
96 simpl3 1059 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
97 eqid 2610 . . . . . . . . . 10 (𝐷𝐼) = (𝐷𝐼)
98 simpl 472 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
99 simprl 790 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
10099elpwid 4118 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
101 simprr 792 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
102101elpwid 4118 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
103100, 102unssd 3751 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
10498, 103sselpwd 4734 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
1051043ad2antl2 1217 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
106 eqid 2610 . . . . . . . . . 10 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10767, 12, 95, 96, 97, 105, 106dssmapfv3d 37333 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
108 simpl1 1057 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝜑)
10967, 12, 13ntrclsfv1 37373 . . . . . . . . . . 11 (𝜑 → (𝐷𝐼) = 𝐾)
110109fveq1d 6105 . . . . . . . . . 10 (𝜑 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
111108, 110syl 17 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
112107, 111eqtr3d 2646 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐾‘(𝑠𝑡)))
113 simprl 790 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
114 eqid 2610 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
11567, 12, 95, 96, 97, 113, 114dssmapfv3d 37333 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
116109fveq1d 6105 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
117108, 116syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
118115, 117eqtr3d 2646 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑠))) = (𝐾𝑠))
119 simprr 792 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
120 eqid 2610 . . . . . . . . . . 11 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
12167, 12, 95, 96, 97, 119, 120dssmapfv3d 37333 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
122109fveq1d 6105 . . . . . . . . . . 11 (𝜑 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
123108, 122syl 17 . . . . . . . . . 10 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
124121, 123eqtr3d 2646 . . . . . . . . 9 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵𝑡))) = (𝐾𝑡))
125118, 124uneq12d 3730 . . . . . . . 8 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) = ((𝐾𝑠) ∪ (𝐾𝑡)))
126112, 125sseq12d 3597 . . . . . . 7 (((𝜑𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12766, 91, 92, 93, 94, 126syl32anc 1326 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12887, 90, 1273bitrd 293 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠𝑡))) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
12957, 65, 1283bitrd 293 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13035, 50, 129ralxfrd2 4810 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13117, 31, 130ralxfrd2 4810 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵((𝐼𝑎) ∩ (𝐼𝑏)) ⊆ (𝐼‘(𝑎𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
13211, 131syl5bb 271 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-frege1 37104 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by: (None)
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