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Theorem ntrclsk13 42417
Description: The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (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
ntrclsk13 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀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 ntrclsk13
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 4170 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
21fveq2d 6851 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
3 fveq2 6847 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
43ineq1d 4176 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
52, 4eqeq12d 2753 . . 3 (𝑠 = 𝑎 → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡))))
6 ineq2 4171 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
76fveq2d 6851 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
8 fveq2 6847 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
98ineq2d 4177 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
107, 9eqeq12d 2753 . . 3 (𝑡 = 𝑏 → ((𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏))))
115, 10cbvral2vw 3230 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
12 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
13 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
1412, 13ntrclsbex 42380 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 4097 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 5288 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 482 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4572 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
1914adantr 482 . . . . . 6 ((𝜑𝑎𝐵) → 𝐵 ∈ V)
20 difssd 4097 . . . . . 6 ((𝜑𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 5288 . . . . 5 ((𝜑𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 difeq2 4081 . . . . . . . 8 (𝑠 = (𝐵𝑎) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2322eqeq2d 2748 . . . . . . 7 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
24 eqcom 2744 . . . . . . 7 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2523, 24bitrdi 287 . . . . . 6 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
2625adantl 483 . . . . 5 (((𝜑𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
27 dfss4 4223 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 215 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 483 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3586 . . . 4 ((𝜑𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3118, 30sylan2 594 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 ineq1 4170 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
3332fveq2d 6851 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
34 fveq2 6847 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
3534ineq1d 4176 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
3633, 35eqeq12d 2753 . . . . . 6 (𝑎 = (𝐵𝑠) → ((𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ (𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
3736ralbidv 3175 . . . . 5 (𝑎 = (𝐵𝑠) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
38373ad2ant3 1136 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
39 difssd 4097 . . . . . . . 8 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
4014, 39sselpwd 5288 . . . . . . 7 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
4140ad2antrr 725 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
42 simpll 766 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝜑)
43 elpwi 4572 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
4443adantl 483 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝑏𝐵)
45 difssd 4097 . . . . . . . . . 10 (𝜑 → (𝐵𝑏) ⊆ 𝐵)
4614, 45sselpwd 5288 . . . . . . . . 9 (𝜑 → (𝐵𝑏) ∈ 𝒫 𝐵)
4746adantr 482 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
48 difeq2 4081 . . . . . . . . . . 11 (𝑡 = (𝐵𝑏) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4948eqeq2d 2748 . . . . . . . . . 10 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
50 eqcom 2744 . . . . . . . . . 10 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5149, 50bitrdi 287 . . . . . . . . 9 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
5251adantl 483 . . . . . . . 8 (((𝜑𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
53 dfss4 4223 . . . . . . . . . 10 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5453biimpi 215 . . . . . . . . 9 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5554adantl 483 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5647, 52, 55rspcedvd 3586 . . . . . . 7 ((𝜑𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5742, 44, 56syl2anc 585 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
58 ineq2 4171 . . . . . . . . . . 11 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
59 difundi 4244 . . . . . . . . . . 11 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6058, 59eqtr4di 2795 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6160fveq2d 6851 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
62 fveq2 6847 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
6362ineq2d 4177 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
6461, 63eqeq12d 2753 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
65643ad2ant3 1136 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
66 simp1l 1198 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
6766, 14jccir 523 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝜑𝐵 ∈ V))
68 simp1r 1199 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
69 simp2 1138 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑡 ∈ 𝒫 𝐵)
70 ntrcls.o . . . . . . . . . . . . . 14 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
7170, 12, 13ntrclsiex 42399 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
72 elmapi 8794 . . . . . . . . . . . . 13 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7371, 72syl 17 . . . . . . . . . . . 12 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
7473anim1i 616 . . . . . . . . . . 11 ((𝜑𝐵 ∈ V) → (𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V))
7574adantr 482 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V))
76 simpl 484 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
77 simpr 486 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → 𝐵 ∈ V)
78 difssd 4097 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
7977, 78sselpwd 5288 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8076, 79ffvelcdmd 7041 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8180elpwid 4574 . . . . . . . . . . 11 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
82 difssd 4097 . . . . . . . . . . . . . . 15 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
8377, 82sselpwd 5288 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
8476, 83ffvelcdmd 7041 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
8584elpwid 4574 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
86 ssinss1 4202 . . . . . . . . . . . 12 ((𝐼‘(𝐵𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
8785, 86syl 17 . . . . . . . . . . 11 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵)
8881, 87jca 513 . . . . . . . . . 10 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵 ∧ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵))
89 rcompleq 4260 . . . . . . . . . 10 (((𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵 ∧ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
9075, 88, 893syl 18 . . . . . . . . 9 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
91 simplr 768 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
9271ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
93 eqid 2737 . . . . . . . . . . 11 (𝐷𝐼) = (𝐷𝐼)
94 simprl 770 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
9594elpwid 4574 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
96 simprr 772 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
9796elpwid 4574 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
9895, 97unssd 4151 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
9991, 98sselpwd 5288 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
100 eqid 2737 . . . . . . . . . . 11 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10170, 12, 91, 92, 93, 99, 100dssmapfv3d 42365 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
102 simpl 484 . . . . . . . . . . . . 13 ((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
103 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
10471ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
105 simpr 486 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
106 eqid 2737 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
10770, 12, 103, 104, 93, 105, 106dssmapfv3d 42365 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
108102, 107sylan2 594 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵𝑠))))
109 simpr 486 . . . . . . . . . . . . 13 ((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
110 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11171ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
112 simpr 486 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
113 eqid 2737 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
11470, 12, 110, 111, 93, 112, 113dssmapfv3d 42365 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
115109, 114sylan2 594 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵𝑡))))
116108, 115uneq12d 4129 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
117 difindi 4246 . . . . . . . . . . 11 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
118116, 117eqtr4di 2795 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
119101, 118eqeq12d 2753 . . . . . . . . 9 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘(𝑠𝑡)) = (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))))
120 simpll 766 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝜑)
12170, 12, 13ntrclsfv1 42401 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
122 fveq1 6846 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
123 fveq1 6846 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
124 fveq1 6846 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
125123, 124uneq12d 4129 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)))
126122, 125eqeq12d 2753 . . . . . . . . . 10 ((𝐷𝐼) = 𝐾 → (((𝐷𝐼)‘(𝑠𝑡)) = (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
127120, 121, 1263syl 18 . . . . . . . . 9 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘(𝑠𝑡)) = (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
12890, 119, 1273bitr2d 307 . . . . . . . 8 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
12967, 68, 69, 128syl12anc 836 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13065, 129bitrd 279 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13141, 57, 130ralxfrd2 5372 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
1321313adant3 1133 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13338, 132bitrd 279 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13417, 31, 133ralxfrd2 5372 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13511, 134bitrid 283 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  cun 3913  cin 3914  wss 3915  𝒫 cpw 4565   class class class wbr 5110  cmpt 5193  wf 6497  cfv 6501  (class class class)co 7362  m cmap 8772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-map 8774
This theorem is referenced by: (None)
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