Theorem isfne4b 36322

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	⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Proof of Theorem isfne4b

Step	Hyp	Ref	Expression
1		isfne.1	. . 3 ⊢ 𝑋 = ∪ 𝐴
2		isfne.2	. . 3 ⊢ 𝑌 = ∪ 𝐵
3	1, 2	isfne4 36321	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
4		simpr 484	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
5	4, 1, 2	3eqtr3g 2787	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 = ∪ 𝐵)
6		uniexg 7696	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐵 ∈ V)
8	5, 7	eqeltrd 2828	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
9		uniexb 7720	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
10	8, 9	sylibr 234	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
11		simpl 482	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐵 ∈ 𝑉)
12		tgss3 22906	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
13	10, 11, 12	syl2anc 584	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
14	13	pm5.32da 579	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))))
15	3, 14	bitr4id 290	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  ∪ cuni 4867   class class class wbr 5102  ‘cfv 6499  topGenctg 17376  Fnecfne 36317

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-topgen 17382  df-fne 36318

This theorem is referenced by:  fnetr  36332  fneval  36333