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Theorem isfne 34528
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 34527 . . . . 5 Rel Fne
21brrelex1i 5643 . . . 4 (𝐴Fne𝐵𝐴 ∈ V)
32anim1i 615 . . 3 ((𝐴Fne𝐵𝐵𝐶) → (𝐴 ∈ V ∧ 𝐵𝐶))
43ancoms 459 . 2 ((𝐵𝐶𝐴Fne𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
5 simpr 485 . . . . 5 ((𝐵𝐶𝑋 = 𝑌) → 𝑋 = 𝑌)
6 isfne.1 . . . . 5 𝑋 = 𝐴
7 isfne.2 . . . . 5 𝑌 = 𝐵
85, 6, 73eqtr3g 2801 . . . 4 ((𝐵𝐶𝑋 = 𝑌) → 𝐴 = 𝐵)
9 simpr 485 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 = 𝐵)
10 uniexg 7593 . . . . . . . 8 (𝐵𝐶 𝐵 ∈ V)
1110adantr 481 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵 ∈ V)
129, 11eqeltrd 2839 . . . . . 6 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
13 uniexb 7614 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1412, 13sylibr 233 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
15 simpl 483 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵𝐶)
1614, 15jca 512 . . . 4 ((𝐵𝐶 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
178, 16syldan 591 . . 3 ((𝐵𝐶𝑋 = 𝑌) → (𝐴 ∈ V ∧ 𝐵𝐶))
1817adantrr 714 . 2 ((𝐵𝐶 ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))) → (𝐴 ∈ V ∧ 𝐵𝐶))
19 unieq 4850 . . . . . 6 (𝑟 = 𝐴 𝑟 = 𝐴)
2019, 6eqtr4di 2796 . . . . 5 (𝑟 = 𝐴 𝑟 = 𝑋)
2120eqeq1d 2740 . . . 4 (𝑟 = 𝐴 → ( 𝑟 = 𝑠𝑋 = 𝑠))
22 raleq 3342 . . . 4 (𝑟 = 𝐴 → (∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥)))
2321, 22anbi12d 631 . . 3 (𝑟 = 𝐴 → (( 𝑟 = 𝑠 ∧ ∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = 𝑠 ∧ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥))))
24 unieq 4850 . . . . . 6 (𝑠 = 𝐵 𝑠 = 𝐵)
2524, 7eqtr4di 2796 . . . . 5 (𝑠 = 𝐵 𝑠 = 𝑌)
2625eqeq2d 2749 . . . 4 (𝑠 = 𝐵 → (𝑋 = 𝑠𝑋 = 𝑌))
27 ineq1 4139 . . . . . . 7 (𝑠 = 𝐵 → (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2827unieqd 4853 . . . . . 6 (𝑠 = 𝐵 (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2928sseq2d 3953 . . . . 5 (𝑠 = 𝐵 → (𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
3029ralbidv 3112 . . . 4 (𝑠 = 𝐵 → (∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥)))
3126, 30anbi12d 631 . . 3 (𝑠 = 𝐵 → ((𝑋 = 𝑠 ∧ ∀𝑥𝐴 𝑥 (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
32 df-fne 34526 . . 3 Fne = {⟨𝑟, 𝑠⟩ ∣ ( 𝑟 = 𝑠 ∧ ∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥))}
3323, 31, 32brabg 5452 . 2 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
344, 18, 33pm5.21nd 799 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533   cuni 4839   class class class wbr 5074  Fnecfne 34525
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-fne 34526
This theorem is referenced by:  isfne4  34529
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