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Theorem isfne3 32675
 Description: The predicate "𝐵 is finer than 𝐴." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem isfne3
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = 𝐴
2 isfne.2 . . 3 𝑌 = 𝐵
31, 2isfne4 32672 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
4 dfss3 3741 . . . 4 (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵))
5 eltg3 20987 . . . . 5 (𝐵𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
65ralbidv 3135 . . . 4 (𝐵𝐶 → (∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦)))
74, 6syl5bb 272 . . 3 (𝐵𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦)))
87anbi2d 614 . 2 (𝐵𝐶 → ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
93, 8syl5bb 272 1 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  ∪ cuni 4575   class class class wbr 4787  ‘cfv 6030  topGenctg 16306  Fnecfne 32668 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-topgen 16312  df-fne 32669 This theorem is referenced by: (None)
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