Theorem fneval 32347

Description: Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
fneval.1	|- .~ = ( Fne i^i `' Fne )

Assertion

Ref	Expression
fneval	|- ( ( A e. V /\ B e. W ) -> ( A .~ B <-> ( topGen ` A ) = ( topGen ` B ) ) )

Proof of Theorem fneval

Step	Hyp	Ref	Expression
1		fneval.1	. . . 4 |- .~ = ( Fne i^i `' Fne )
2	1	breqi 4659	. . 3 |- ( A .~ B <-> A ( Fne i^i `' Fne ) B )
3		brin 4704	. . . 4 |- ( A ( Fne i^i `' Fne ) B <-> ( A Fne B /\ A `' Fne B ) )
4		fnerel 32333	. . . . . 6 |- Rel Fne
5	4	relbrcnv 5506	. . . . 5 |- ( A `' Fne B <-> B Fne A )
6	5	anbi2i 730	. . . 4 |- ( ( A Fne B /\ A `' Fne B ) <-> ( A Fne B /\ B Fne A ) )
7	3, 6	bitri 264	. . 3 |- ( A ( Fne i^i `' Fne ) B <-> ( A Fne B /\ B Fne A ) )
8	2, 7	bitri 264	. 2 |- ( A .~ B <-> ( A Fne B /\ B Fne A ) )
9		eqid 2622	. . . . . 6 |- U. A = U. A
10		eqid 2622	. . . . . 6 |- U. B = U. B
11	9, 10	isfne4b 32336	. . . . 5 |- ( B e. W -> ( A Fne B <-> ( U. A = U. B /\ ( topGen ` A ) C_ ( topGen ` B ) ) ) )
12	10, 9	isfne4b 32336	. . . . . 6 |- ( A e. V -> ( B Fne A <-> ( U. B = U. A /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) )
13		eqcom 2629	. . . . . . 7 |- ( U. B = U. A <-> U. A = U. B )
14	13	anbi1i 731	. . . . . 6 |- ( ( U. B = U. A /\ ( topGen ` B ) C_ ( topGen ` A ) ) <-> ( U. A = U. B /\ ( topGen ` B ) C_ ( topGen ` A ) ) )
15	12, 14	syl6bb 276	. . . . 5 |- ( A e. V -> ( B Fne A <-> ( U. A = U. B /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) )
16	11, 15	bi2anan9r 918	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A Fne B /\ B Fne A ) <-> ( ( U. A = U. B /\ ( topGen ` A ) C_ ( topGen ` B ) ) /\ ( U. A = U. B /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) ) )
17		eqss 3618	. . . . . 6 |- ( ( topGen ` A ) = ( topGen ` B ) <-> ( ( topGen ` A ) C_ ( topGen ` B ) /\ ( topGen ` B ) C_ ( topGen ` A ) ) )
18	17	anbi2i 730	. . . . 5 |- ( ( U. A = U. B /\ ( topGen ` A ) = ( topGen ` B ) ) <-> ( U. A = U. B /\ ( ( topGen ` A ) C_ ( topGen ` B ) /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) )
19		anandi 871	. . . . 5 |- ( ( U. A = U. B /\ ( ( topGen ` A ) C_ ( topGen ` B ) /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) <-> ( ( U. A = U. B /\ ( topGen ` A ) C_ ( topGen ` B ) ) /\ ( U. A = U. B /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) )
20	18, 19	bitri 264	. . . 4 |- ( ( U. A = U. B /\ ( topGen ` A ) = ( topGen ` B ) ) <-> ( ( U. A = U. B /\ ( topGen ` A ) C_ ( topGen ` B ) ) /\ ( U. A = U. B /\ ( topGen ` B ) C_ ( topGen ` A ) ) ) )
21	16, 20	syl6bbr 278	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A Fne B /\ B Fne A ) <-> ( U. A = U. B /\ ( topGen ` A ) = ( topGen ` B ) ) ) )
22		unieq 4444	. . . . 5 |- ( ( topGen ` A ) = ( topGen ` B ) -> U. ( topGen ` A ) = U. ( topGen ` B ) )
23		unitg 20771	. . . . . 6 |- ( A e. V -> U. ( topGen ` A ) = U. A )
24		unitg 20771	. . . . . 6 |- ( B e. W -> U. ( topGen ` B ) = U. B )
25	23, 24	eqeqan12d 2638	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( U. ( topGen ` A ) = U. ( topGen ` B ) <-> U. A = U. B ) )
26	22, 25	syl5ib 234	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( topGen ` A ) = ( topGen ` B ) -> U. A = U. B ) )
27	26	pm4.71rd 667	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( topGen ` A ) = ( topGen ` B ) <-> ( U. A = U. B /\ ( topGen ` A ) = ( topGen ` B ) ) ) )
28	21, 27	bitr4d 271	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A Fne B /\ B Fne A ) <-> ( topGen ` A ) = ( topGen ` B ) ) )
29	8, 28	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( A .~ B <-> ( topGen ` A ) = ( topGen ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   ` cfv 5888   topGenctg 16098   Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-fne 32332

This theorem is referenced by:  fneer  32348  topfneec  32350  topfneec2  32351