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Theorem isfne4 32030
Description: The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne4 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))

Proof of Theorem isfne4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnerel 32028 . . 3 Rel Fne
21brrelex2i 5129 . 2 (𝐴Fne𝐵𝐵 ∈ V)
3 simpl 473 . . . . 5 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝑋 = 𝑌)
4 isfne.1 . . . . 5 𝑋 = 𝐴
5 isfne.2 . . . . 5 𝑌 = 𝐵
63, 4, 53eqtr3g 2678 . . . 4 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝐴 = 𝐵)
7 fvex 6168 . . . . . . 7 (topGen‘𝐵) ∈ V
87ssex 4772 . . . . . 6 (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V)
98adantl 482 . . . . 5 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V)
10 uniexb 6936 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
119, 10sylib 208 . . . 4 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V)
126, 11eqeltrrd 2699 . . 3 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
13 uniexb 6936 . . 3 (𝐵 ∈ V ↔ 𝐵 ∈ V)
1412, 13sylibr 224 . 2 ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
154, 5isfne 32029 . . 3 (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
16 dfss3 3578 . . . . 5 (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵))
17 eltg 20701 . . . . . 6 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
1817ralbidv 2982 . . . . 5 (𝐵 ∈ V → (∀𝑥𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥)))
1916, 18syl5bb 272 . . . 4 (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥)))
2019anbi2d 739 . . 3 (𝐵 ∈ V → ((𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
2115, 20bitr4d 271 . 2 (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
222, 14, 21pm5.21nii 368 1 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cin 3559  wss 3560  𝒫 cpw 4136   cuni 4409   class class class wbr 4623  cfv 5857  topGenctg 16038  Fnecfne 32026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-topgen 16044  df-fne 32027
This theorem is referenced by:  isfne4b  32031  isfne2  32032  isfne3  32033  fnebas  32034  fnetg  32035  topfne  32044  fnemeet1  32056  fnemeet2  32057  fnejoin1  32058  fnejoin2  32059
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