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Theorem ntrclsk13 42822
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 4206 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
21fveq2d 6896 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
3 fveq2 6892 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
43ineq1d 4212 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
52, 4eqeq12d 2749 . . 3 (𝑠 = 𝑎 → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡))))
6 ineq2 4207 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
76fveq2d 6896 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
8 fveq2 6892 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
98ineq2d 4213 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
107, 9eqeq12d 2749 . . 3 (𝑡 = 𝑏 → ((𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏))))
115, 10cbvral2vw 3239 . 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 42785 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 4133 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 5327 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 482 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4610 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
1914adantr 482 . . . . . 6 ((𝜑𝑎𝐵) → 𝐵 ∈ V)
20 difssd 4133 . . . . . 6 ((𝜑𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 5327 . . . . 5 ((𝜑𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 difeq2 4117 . . . . . . . 8 (𝑠 = (𝐵𝑎) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2322eqeq2d 2744 . . . . . . 7 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
24 eqcom 2740 . . . . . . 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 4259 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 215 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 483 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3615 . . . 4 ((𝜑𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3118, 30sylan2 594 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 ineq1 4206 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
3332fveq2d 6896 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
34 fveq2 6892 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
3534ineq1d 4212 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
3633, 35eqeq12d 2749 . . . . . 6 (𝑎 = (𝐵𝑠) → ((𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ (𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
3736ralbidv 3178 . . . . 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 4133 . . . . . . . 8 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
4014, 39sselpwd 5327 . . . . . . 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 4610 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
4443adantl 483 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝑏𝐵)
45 difssd 4133 . . . . . . . . . 10 (𝜑 → (𝐵𝑏) ⊆ 𝐵)
4614, 45sselpwd 5327 . . . . . . . . 9 (𝜑 → (𝐵𝑏) ∈ 𝒫 𝐵)
4746adantr 482 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
48 difeq2 4117 . . . . . . . . . . 11 (𝑡 = (𝐵𝑏) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4948eqeq2d 2744 . . . . . . . . . 10 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
50 eqcom 2740 . . . . . . . . . 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 4259 . . . . . . . . . 10 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5453biimpi 215 . . . . . . . . 9 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5554adantl 483 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5647, 52, 55rspcedvd 3615 . . . . . . 7 ((𝜑𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5742, 44, 56syl2anc 585 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
58 ineq2 4207 . . . . . . . . . . 11 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
59 difundi 4280 . . . . . . . . . . 11 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6058, 59eqtr4di 2791 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6160fveq2d 6896 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
62 fveq2 6892 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
6362ineq2d 4213 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
6461, 63eqeq12d 2749 . . . . . . . 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 42804 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
72 elmapi 8843 . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
7977, 78sselpwd 5327 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8076, 79ffvelcdmd 7088 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8180elpwid 4612 . . . . . . . . . . 11 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
82 difssd 4133 . . . . . . . . . . . . . . 15 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
8377, 82sselpwd 5327 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
8476, 83ffvelcdmd 7088 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
8584elpwid 4612 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
86 ssinss1 4238 . . . . . . . . . . . 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 4296 . . . . . . . . . 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 2733 . . . . . . . . . . 11 (𝐷𝐼) = (𝐷𝐼)
94 simprl 770 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
9594elpwid 4612 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
96 simprr 772 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
9796elpwid 4612 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
9895, 97unssd 4187 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
9991, 98sselpwd 5327 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
100 eqid 2733 . . . . . . . . . . 11 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10170, 12, 91, 92, 93, 99, 100dssmapfv3d 42770 . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
10770, 12, 103, 104, 93, 105, 106dssmapfv3d 42770 . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
11470, 12, 110, 111, 93, 112, 113dssmapfv3d 42770 . . . . . . . . . . . . 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 4165 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
117 difindi 4282 . . . . . . . . . . 11 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
118116, 117eqtr4di 2791 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
119101, 118eqeq12d 2749 . . . . . . . . 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 42806 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
122 fveq1 6891 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
123 fveq1 6891 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
124 fveq1 6891 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
125123, 124uneq12d 4165 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)))
126122, 125eqeq12d 2749 . . . . . . . . . 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 5411 . . . . 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 5411 . 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 3062  wrex 3071  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  𝒫 cpw 4603   class class class wbr 5149  cmpt 5232  wf 6540  cfv 6544  (class class class)co 7409  m cmap 8820
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by: (None)
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