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

Proof of Theorem isfne4b

Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
2		isfne.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
3		isfne.2	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐵
4	1, 2, 3	3eqtr3g 2854	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 = ∪ 𝐵)
5		uniexg 7325	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
6	5	adantr 481	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐵 ∈ V)
7	4, 6	eqeltrd 2883	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
8		uniexb 7343	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
9	7, 8	sylibr 235	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
10		simpl 483	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐵 ∈ 𝑉)
11		tgss3 21278	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
12	9, 10, 11	syl2anc 584	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
13	12	pm5.32da 579	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))))
14	2, 3	isfne4 33298	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
15	13, 14	syl6rbbr 291	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

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