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Theorem isfne 36340
Description: The predicate "𝐵 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hints:   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem isfne
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 36339 . . . . 5 Rel Fne
21brrelex1i 5741 . . . 4 (𝐴Fne𝐵𝐴 ∈ V)
32anim1i 615 . . 3 ((𝐴Fne𝐵𝐵𝐶) → (𝐴 ∈ V ∧ 𝐵𝐶))
43ancoms 458 . 2 ((𝐵𝐶𝐴Fne𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
5 simpr 484 . . . . 5 ((𝐵𝐶𝑋 = 𝑌) → 𝑋 = 𝑌)
6 isfne.1 . . . . 5 𝑋 = 𝐴
7 isfne.2 . . . . 5 𝑌 = 𝐵
85, 6, 73eqtr3g 2800 . . . 4 ((𝐵𝐶𝑋 = 𝑌) → 𝐴 = 𝐵)
9 simpr 484 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 = 𝐵)
10 uniexg 7760 . . . . . . . 8 (𝐵𝐶 𝐵 ∈ V)
1110adantr 480 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵 ∈ V)
129, 11eqeltrd 2841 . . . . . 6 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
13 uniexb 7784 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1412, 13sylibr 234 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
15 simpl 482 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵𝐶)
1614, 15jca 511 . . . 4 ((𝐵𝐶 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
178, 16syldan 591 . . 3 ((𝐵𝐶𝑋 = 𝑌) → (𝐴 ∈ V ∧ 𝐵𝐶))
1817adantrr 717 . 2 ((𝐵𝐶 ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))) → (𝐴 ∈ V ∧ 𝐵𝐶))
19 unieq 4918 . . . . . 6 (𝑟 = 𝐴 𝑟 = 𝐴)
2019, 6eqtr4di 2795 . . . . 5 (𝑟 = 𝐴 𝑟 = 𝑋)
2120eqeq1d 2739 . . . 4 (𝑟 = 𝐴 → ( 𝑟 = 𝑠𝑋 = 𝑠))
22 raleq 3323 . . . 4 (𝑟 = 𝐴 → (∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥)))
2321, 22anbi12d 632 . . 3 (𝑟 = 𝐴 → (( 𝑟 = 𝑠 ∧ ∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = 𝑠 ∧ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥))))
24 unieq 4918 . . . . . 6 (𝑠 = 𝐵 𝑠 = 𝐵)
2524, 7eqtr4di 2795 . . . . 5 (𝑠 = 𝐵 𝑠 = 𝑌)
2625eqeq2d 2748 . . . 4 (𝑠 = 𝐵 → (𝑋 = 𝑠𝑋 = 𝑌))
27 ineq1 4213 . . . . . . 7 (𝑠 = 𝐵 → (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2827unieqd 4920 . . . . . 6 (𝑠 = 𝐵 (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2928sseq2d 4016 . . . . 5 (𝑠 = 𝐵 → (𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
3029ralbidv 3178 . . . 4 (𝑠 = 𝐵 → (∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥)))
3126, 30anbi12d 632 . . 3 (𝑠 = 𝐵 → ((𝑋 = 𝑠 ∧ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
32 df-fne 36338 . . 3 Fne = {⟨𝑟, 𝑠⟩ ∣ ( 𝑟 = 𝑠 ∧ ∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥))}
3323, 31, 32brabg 5544 . 2 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
344, 18, 33pm5.21nd 802 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  Fnecfne 36337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-fne 36338
This theorem is referenced by:  isfne4  36341
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