Theorem isfne4 36306

Description: The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
isfne.1	⊢ 𝑋 = ∪ 𝐴
isfne.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
isfne4	⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))

Proof of Theorem isfne4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnerel 36304	. . 3 ⊢ Rel Fne
2	1	brrelex2i 5757	. 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V)
3		simpl 482	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝑋 = 𝑌)
4		isfne.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐴
5		isfne.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐵
6	3, 4, 5	3eqtr3g 2803	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵)
7		fvex 6933	. . . . . . 7 ⊢ (topGen‘𝐵) ∈ V
8	7	ssex 5339	. . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V)
9	8	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V)
10	9	uniexd 7777	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V)
11	6, 10	eqeltrrd 2845	. . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V)
12		uniexb 7799	. . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
13	11, 12	sylibr 234	. 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
14	4, 5	isfne 36305	. . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))
15		dfss3 3997	. . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵))
16		eltg 22985	. . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
17	16	ralbidv 3184	. . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
18	15, 17	bitrid 283	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
19	18	anbi2d 629	. . 3 ⊢ (𝐵 ∈ V → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))
20	14, 19	bitr4d 282	. 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))))
21	2, 13, 20	pm5.21nii 378	1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  ‘cfv 6573  topGenctg 17497  Fnecfne 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topgen 17503  df-fne 36303

This theorem is referenced by:  isfne4b  36307  isfne2  36308  isfne3  36309  fnebas  36310  fnetg  36311  topfne  36320  fnemeet1  36332  fnemeet2  36333  fnejoin1  36334  fnejoin2  36335