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Theorem isfne2 33694
Description: The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne2 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧
Allowed substitution hints:   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem isfne2
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = 𝐴
2 isfne.2 . . 3 𝑌 = 𝐵
31, 2isfne4 33692 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
4 dfss3 3959 . . . 4 (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵))
5 eltg2b 21570 . . . . 5 (𝐵𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
65ralbidv 3200 . . . 4 (𝐵𝐶 → (∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
74, 6syl5bb 285 . . 3 (𝐵𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
87anbi2d 630 . 2 (𝐵𝐶 → ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
93, 8syl5bb 285 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  wral 3141  wrex 3142  wss 3939   cuni 4841   class class class wbr 5069  cfv 6358  topGenctg 16714  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-sbc 3776  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-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-topgen 16720  df-fne 33689
This theorem is referenced by:  fness  33701  fneref  33702  fnessref  33709
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