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Theorem isfne4b 33299
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 485 . . . . . . 7 ((𝐵𝑉𝑋 = 𝑌) → 𝑋 = 𝑌)
2 isfne.1 . . . . . . 7 𝑋 = 𝐴
3 isfne.2 . . . . . . 7 𝑌 = 𝐵
41, 2, 33eqtr3g 2854 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 = 𝐵)
5 uniexg 7325 . . . . . . 7 (𝐵𝑉 𝐵 ∈ V)
65adantr 481 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐵 ∈ V)
74, 6eqeltrd 2883 . . . . 5 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
8 uniexb 7343 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8sylibr 235 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
10 simpl 483 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐵𝑉)
11 tgss3 21278 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
129, 10, 11syl2anc 584 . . 3 ((𝐵𝑉𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1312pm5.32da 579 . 2 (𝐵𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
142, 3isfne4 33298 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
1513, 14syl6rbbr 291 1 (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  wss 3859   cuni 4745   class class class wbr 4962  cfv 6225  topGenctg 16540  Fnecfne 33294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-topgen 16546  df-fne 33295
This theorem is referenced by:  fnetr  33309  fneval  33310
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