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Theorem ntrclsk13 40414
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 4180 . . . . 5 (𝑠 = 𝑎 → (𝑠𝑡) = (𝑎𝑡))
21fveq2d 6668 . . . 4 (𝑠 = 𝑎 → (𝐼‘(𝑠𝑡)) = (𝐼‘(𝑎𝑡)))
3 fveq2 6664 . . . . 5 (𝑠 = 𝑎 → (𝐼𝑠) = (𝐼𝑎))
43ineq1d 4187 . . . 4 (𝑠 = 𝑎 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)))
52, 4eqeq12d 2837 . . 3 (𝑠 = 𝑎 → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡))))
6 ineq2 4182 . . . . 5 (𝑡 = 𝑏 → (𝑎𝑡) = (𝑎𝑏))
76fveq2d 6668 . . . 4 (𝑡 = 𝑏 → (𝐼‘(𝑎𝑡)) = (𝐼‘(𝑎𝑏)))
8 fveq2 6664 . . . . 5 (𝑡 = 𝑏 → (𝐼𝑡) = (𝐼𝑏))
98ineq2d 4188 . . . 4 (𝑡 = 𝑏 → ((𝐼𝑎) ∩ (𝐼𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑏)))
107, 9eqeq12d 2837 . . 3 (𝑡 = 𝑏 → ((𝐼‘(𝑎𝑡)) = ((𝐼𝑎) ∩ (𝐼𝑡)) ↔ (𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏))))
115, 10cbvral2vw 3461 . 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 40377 . . . . 5 (𝜑𝐵 ∈ V)
15 difssd 4108 . . . . 5 (𝜑 → (𝐵𝑠) ⊆ 𝐵)
1614, 15sselpwd 5222 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
1716adantr 483 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
18 elpwi 4550 . . . 4 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
1914adantr 483 . . . . . 6 ((𝜑𝑎𝐵) → 𝐵 ∈ V)
20 difssd 4108 . . . . . 6 ((𝜑𝑎𝐵) → (𝐵𝑎) ⊆ 𝐵)
2119, 20sselpwd 5222 . . . . 5 ((𝜑𝑎𝐵) → (𝐵𝑎) ∈ 𝒫 𝐵)
22 difeq2 4092 . . . . . . . 8 (𝑠 = (𝐵𝑎) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑎)))
2322eqeq2d 2832 . . . . . . 7 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵𝑎))))
24 eqcom 2828 . . . . . . 7 (𝑎 = (𝐵 ∖ (𝐵𝑎)) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2523, 24syl6bb 289 . . . . . 6 (𝑠 = (𝐵𝑎) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
2625adantl 484 . . . . 5 (((𝜑𝑎𝐵) ∧ 𝑠 = (𝐵𝑎)) → (𝑎 = (𝐵𝑠) ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎))
27 dfss4 4234 . . . . . . 7 (𝑎𝐵 ↔ (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2827biimpi 218 . . . . . 6 (𝑎𝐵 → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
2928adantl 484 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 ∖ (𝐵𝑎)) = 𝑎)
3021, 26, 29rspcedvd 3625 . . . 4 ((𝜑𝑎𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
3118, 30sylan2 594 . . 3 ((𝜑𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠))
32 ineq1 4180 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝑎𝑏) = ((𝐵𝑠) ∩ 𝑏))
3332fveq2d 6668 . . . . . . 7 (𝑎 = (𝐵𝑠) → (𝐼‘(𝑎𝑏)) = (𝐼‘((𝐵𝑠) ∩ 𝑏)))
34 fveq2 6664 . . . . . . . 8 (𝑎 = (𝐵𝑠) → (𝐼𝑎) = (𝐼‘(𝐵𝑠)))
3534ineq1d 4187 . . . . . . 7 (𝑎 = (𝐵𝑠) → ((𝐼𝑎) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)))
3633, 35eqeq12d 2837 . . . . . 6 (𝑎 = (𝐵𝑠) → ((𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ (𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
3736ralbidv 3197 . . . . 5 (𝑎 = (𝐵𝑠) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
38373ad2ant3 1131 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏))))
39 difssd 4108 . . . . . . . 8 (𝜑 → (𝐵𝑡) ⊆ 𝐵)
4014, 39sselpwd 5222 . . . . . . 7 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
4140ad2antrr 724 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
42 simpll 765 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝜑)
43 elpwi 4550 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
4443adantl 484 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝑏𝐵)
45 difssd 4108 . . . . . . . . . 10 (𝜑 → (𝐵𝑏) ⊆ 𝐵)
4614, 45sselpwd 5222 . . . . . . . . 9 (𝜑 → (𝐵𝑏) ∈ 𝒫 𝐵)
4746adantr 483 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵𝑏) ∈ 𝒫 𝐵)
48 difeq2 4092 . . . . . . . . . . 11 (𝑡 = (𝐵𝑏) → (𝐵𝑡) = (𝐵 ∖ (𝐵𝑏)))
4948eqeq2d 2832 . . . . . . . . . 10 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵𝑏))))
50 eqcom 2828 . . . . . . . . . 10 (𝑏 = (𝐵 ∖ (𝐵𝑏)) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5149, 50syl6bb 289 . . . . . . . . 9 (𝑡 = (𝐵𝑏) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
5251adantl 484 . . . . . . . 8 (((𝜑𝑏𝐵) ∧ 𝑡 = (𝐵𝑏)) → (𝑏 = (𝐵𝑡) ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏))
53 dfss4 4234 . . . . . . . . . 10 (𝑏𝐵 ↔ (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5453biimpi 218 . . . . . . . . 9 (𝑏𝐵 → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5554adantl 484 . . . . . . . 8 ((𝜑𝑏𝐵) → (𝐵 ∖ (𝐵𝑏)) = 𝑏)
5647, 52, 55rspcedvd 3625 . . . . . . 7 ((𝜑𝑏𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
5742, 44, 56syl2anc 586 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡))
58 ineq2 4182 . . . . . . . . . . 11 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = ((𝐵𝑠) ∩ (𝐵𝑡)))
59 difundi 4255 . . . . . . . . . . 11 (𝐵 ∖ (𝑠𝑡)) = ((𝐵𝑠) ∩ (𝐵𝑡))
6058, 59syl6eqr 2874 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → ((𝐵𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠𝑡)))
6160fveq2d 6668 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → (𝐼‘((𝐵𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠𝑡))))
62 fveq2 6664 . . . . . . . . . 10 (𝑏 = (𝐵𝑡) → (𝐼𝑏) = (𝐼‘(𝐵𝑡)))
6362ineq2d 4188 . . . . . . . . 9 (𝑏 = (𝐵𝑡) → ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))))
6461, 63eqeq12d 2837 . . . . . . . 8 (𝑏 = (𝐵𝑡) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
65643ad2ant3 1131 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
66 simp1l 1193 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝜑)
6766, 14jccir 524 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → (𝜑𝐵 ∈ V))
68 simp1r 1194 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → 𝑠 ∈ 𝒫 𝐵)
69 simp2 1133 . . . . . . . 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 40396 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
72 elmapi 8422 . . . . . . . . . . . . 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 483 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V))
76 simpl 485 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
77 simpr 487 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → 𝐵 ∈ V)
78 difssd 4108 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ⊆ 𝐵)
7977, 78sselpwd 5222 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵 ∖ (𝑠𝑡)) ∈ 𝒫 𝐵)
8076, 79ffvelrnd 6846 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ∈ 𝒫 𝐵)
8180elpwid 4552 . . . . . . . . . . 11 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵)
82 difssd 4108 . . . . . . . . . . . . . . 15 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ⊆ 𝐵)
8377, 82sselpwd 5222 . . . . . . . . . . . . . 14 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐵𝑠) ∈ 𝒫 𝐵)
8476, 83ffvelrnd 6846 . . . . . . . . . . . . 13 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ∈ 𝒫 𝐵)
8584elpwid 4552 . . . . . . . . . . . 12 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → (𝐼‘(𝐵𝑠)) ⊆ 𝐵)
86 ssinss1 4213 . . . . . . . . . . . 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 514 . . . . . . . . . 10 ((𝐼:𝒫 𝐵⟶𝒫 𝐵𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) ⊆ 𝐵 ∧ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ⊆ 𝐵))
89 rcompleq 40363 . . . . . . . . . 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 767 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
9271ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
93 eqid 2821 . . . . . . . . . . 11 (𝐷𝐼) = (𝐷𝐼)
94 simprl 769 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
9594elpwid 4552 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑠𝐵)
96 simprr 771 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
9796elpwid 4552 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝑡𝐵)
9895, 97unssd 4161 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ⊆ 𝐵)
9991, 98sselpwd 5222 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (𝑠𝑡) ∈ 𝒫 𝐵)
100 eqid 2821 . . . . . . . . . . 11 ((𝐷𝐼)‘(𝑠𝑡)) = ((𝐷𝐼)‘(𝑠𝑡))
10170, 12, 91, 92, 93, 99, 100dssmapfv3d 40358 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠𝑡)))))
102 simpl 485 . . . . . . . . . . . . 13 ((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
103 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
10471ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
105 simpr 487 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
106 eqid 2821 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑠) = ((𝐷𝐼)‘𝑠)
10770, 12, 103, 104, 93, 105, 106dssmapfv3d 40358 . . . . . . . . . . . . 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 487 . . . . . . . . . . . . 13 ((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
110 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11171ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
112 simpr 487 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
113 eqid 2821 . . . . . . . . . . . . . 14 ((𝐷𝐼)‘𝑡) = ((𝐷𝐼)‘𝑡)
11470, 12, 110, 111, 93, 112, 113dssmapfv3d 40358 . . . . . . . . . . . . 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 4139 . . . . . . . . . . 11 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡)))))
117 difindi 4257 . . . . . . . . . . 11 (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵𝑡))))
118116, 117syl6eqr 2874 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = (𝐵 ∖ ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡)))))
119101, 118eqeq12d 2837 . . . . . . . . 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 765 . . . . . . . . . 10 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → 𝜑)
12170, 12, 13ntrclsfv1 40398 . . . . . . . . . 10 (𝜑 → (𝐷𝐼) = 𝐾)
122 fveq1 6663 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘(𝑠𝑡)) = (𝐾‘(𝑠𝑡)))
123 fveq1 6663 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑠) = (𝐾𝑠))
124 fveq1 6663 . . . . . . . . . . . 12 ((𝐷𝐼) = 𝐾 → ((𝐷𝐼)‘𝑡) = (𝐾𝑡))
125123, 124uneq12d 4139 . . . . . . . . . . 11 ((𝐷𝐼) = 𝐾 → (((𝐷𝐼)‘𝑠) ∪ ((𝐷𝐼)‘𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)))
126122, 125eqeq12d 2837 . . . . . . . . . 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 309 . . . . . . . 8 (((𝜑𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵)) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
12967, 68, 69, 128syl12anc 834 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘(𝐵 ∖ (𝑠𝑡))) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼‘(𝐵𝑡))) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13065, 129bitrd 281 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵𝑡)) → ((𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13141, 57, 130ralxfrd2 5304 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
1321313adant3 1128 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵𝑠)) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13338, 132bitrd 281 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13417, 31, 133ralxfrd2 5304 . 2 (𝜑 → (∀𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎𝑏)) = ((𝐼𝑎) ∩ (𝐼𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
13511, 134syl5bb 285 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 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3494  cdif 3932  cun 3933  cin 3934  wss 3935  𝒫 cpw 4538   class class class wbr 5058  cmpt 5138  wf 6345  cfv 6349  (class class class)co 7150  m cmap 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402
This theorem is referenced by: (None)
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