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Theorem isfne4b 32031
Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne4b (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Proof of Theorem isfne4b
StepHypRef Expression
1 simpr 477 . . . . . . 7 ((𝐵𝑉𝑋 = 𝑌) → 𝑋 = 𝑌)
2 isfne.1 . . . . . . 7 𝑋 = 𝐴
3 isfne.2 . . . . . . 7 𝑌 = 𝐵
41, 2, 33eqtr3g 2678 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 = 𝐵)
5 uniexg 6920 . . . . . . 7 (𝐵𝑉 𝐵 ∈ V)
65adantr 481 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐵 ∈ V)
74, 6eqeltrd 2698 . . . . 5 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
8 uniexb 6936 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8sylibr 224 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
10 simpl 473 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐵𝑉)
11 tgss3 20730 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
129, 10, 11syl2anc 692 . . 3 ((𝐵𝑉𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1312pm5.32da 672 . 2 (𝐵𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
142, 3isfne4 32030 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
1513, 14syl6rbbr 279 1 (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560   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-ne 2791  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-iun 4494  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:  fnetr  32041  fneval  32042
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