Theorem isfne4b 36872

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 36871	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
4		simpr 489	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
5	4, 1, 2	3eqtr3g 2821	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 = ∪ 𝐵)
6		uniexg 7738	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
7	6	adantr 485	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐵 ∈ V)
8	5, 7	eqeltrd 2863	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
9		uniexb 7759	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
10	8, 9	sylibr 237	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
11		simpl 487	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐵 ∈ 𝑉)
12		tgss3 23143	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
13	10, 11, 12	syl2anc 595	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
14	13	pm5.32da 589	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))))
15	3, 14	bitr4id 293	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4872   class class class wbr 5109  ‘cfv 6536  topGenctg 17485  Fnecfne 36867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-topgen 17491  df-fne 36868

This theorem is referenced by:  fnetr  36882  fneval  36883