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Theorem isfne 33691
 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 33690 . . . . 5 Rel Fne
21brrelex1i 5611 . . . 4 (𝐴Fne𝐵𝐴 ∈ V)
32anim1i 616 . . 3 ((𝐴Fne𝐵𝐵𝐶) → (𝐴 ∈ V ∧ 𝐵𝐶))
43ancoms 461 . 2 ((𝐵𝐶𝐴Fne𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
5 simpr 487 . . . . 5 ((𝐵𝐶𝑋 = 𝑌) → 𝑋 = 𝑌)
6 isfne.1 . . . . 5 𝑋 = 𝐴
7 isfne.2 . . . . 5 𝑌 = 𝐵
85, 6, 73eqtr3g 2882 . . . 4 ((𝐵𝐶𝑋 = 𝑌) → 𝐴 = 𝐵)
9 simpr 487 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 = 𝐵)
10 uniexg 7469 . . . . . . . 8 (𝐵𝐶 𝐵 ∈ V)
1110adantr 483 . . . . . . 7 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵 ∈ V)
129, 11eqeltrd 2916 . . . . . 6 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
13 uniexb 7489 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1412, 13sylibr 236 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐴 ∈ V)
15 simpl 485 . . . . 5 ((𝐵𝐶 𝐴 = 𝐵) → 𝐵𝐶)
1614, 15jca 514 . . . 4 ((𝐵𝐶 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝐶))
178, 16syldan 593 . . 3 ((𝐵𝐶𝑋 = 𝑌) → (𝐴 ∈ V ∧ 𝐵𝐶))
1817adantrr 715 . 2 ((𝐵𝐶 ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))) → (𝐴 ∈ V ∧ 𝐵𝐶))
19 unieq 4852 . . . . . 6 (𝑟 = 𝐴 𝑟 = 𝐴)
2019, 6syl6eqr 2877 . . . . 5 (𝑟 = 𝐴 𝑟 = 𝑋)
2120eqeq1d 2826 . . . 4 (𝑟 = 𝐴 → ( 𝑟 = 𝑠𝑋 = 𝑠))
22 raleq 3408 . . . 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 4852 . . . . . 6 (𝑠 = 𝐵 𝑠 = 𝐵)
2524, 7syl6eqr 2877 . . . . 5 (𝑠 = 𝐵 𝑠 = 𝑌)
2625eqeq2d 2835 . . . 4 (𝑠 = 𝐵 → (𝑋 = 𝑠𝑋 = 𝑌))
27 ineq1 4184 . . . . . . 7 (𝑠 = 𝐵 → (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2827unieqd 4855 . . . . . 6 (𝑠 = 𝐵 (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
2928sseq2d 4002 . . . . 5 (𝑠 = 𝐵 → (𝑥 (𝑠 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
3029ralbidv 3200 . . . 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 33689 . . 3 Fne = {⟨𝑟, 𝑠⟩ ∣ ( 𝑟 = 𝑠 ∧ ∀𝑥𝑟 𝑥 (𝑠 ∩ 𝒫 𝑥))}
3323, 31, 32brabg 5429 . 2 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
344, 18, 33pm5.21nd 800 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841   class class class wbr 5069  Fnecfne 33688 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-fne 33689 This theorem is referenced by:  isfne4  33692
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