Theorem nfvres 5462

Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
nfvres	|- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )

Proof of Theorem nfvres

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5139	. . . . . . . . . 10 |- ( ( F |` B ) ` A ) = ( iota x A ( F |` B ) x )
2		df-iota 5096	. . . . . . . . . 10 |- ( iota x A ( F |` B ) x ) = U. { y | { x | A ( F |` B ) x } = { y } }
3	1, 2	eqtri 2161	. . . . . . . . 9 |- ( ( F |` B ) ` A ) = U. { y | { x | A ( F |` B ) x } = { y } }
4	3	eleq2i 2207	. . . . . . . 8 |- ( z e. ( ( F |` B ) ` A ) <-> z e. U. { y | { x | A ( F |` B ) x } = { y } } )
5		eluni 3747	. . . . . . . 8 |- ( z e. U. { y | { x | A ( F |` B ) x } = { y } } <-> E. w ( z e. w /\ w e. { y | { x | A ( F |` B ) x } = { y } } ) )
6	4, 5	bitri 183	. . . . . . 7 |- ( z e. ( ( F |` B ) ` A ) <-> E. w ( z e. w /\ w e. { y | { x | A ( F |` B ) x } = { y } } ) )
7		exsimpr 1598	. . . . . . 7 |- ( E. w ( z e. w /\ w e. { y | { x | A ( F |` B ) x } = { y } } ) -> E. w w e. { y | { x | A ( F |` B ) x } = { y } } )
8	6, 7	sylbi 120	. . . . . 6 |- ( z e. ( ( F |` B ) ` A ) -> E. w w e. { y | { x | A ( F |` B ) x } = { y } } )
9		df-clab 2127	. . . . . . . 8 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> [ w / y ] { x | A ( F |` B ) x } = { y } )
10		nfv 1509	. . . . . . . . 9 |- F/ y { x | A ( F |` B ) x } = { w }
11		sneq 3543	. . . . . . . . . 10 |- ( y = w -> { y } = { w } )
12	11	eqeq2d 2152	. . . . . . . . 9 |- ( y = w -> ( { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } ) )
13	10, 12	sbie 1765	. . . . . . . 8 |- ( [ w / y ] { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } )
14	9, 13	bitri 183	. . . . . . 7 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> { x | A ( F |` B ) x } = { w } )
15	14	exbii 1585	. . . . . 6 |- ( E. w w e. { y | { x | A ( F |` B ) x } = { y } } <-> E. w { x | A ( F |` B ) x } = { w } )
16	8, 15	sylib 121	. . . . 5 |- ( z e. ( ( F |` B ) ` A ) -> E. w { x | A ( F |` B ) x } = { w } )
17		euabsn2 3600	. . . . 5 |- ( E! x A ( F |` B ) x <-> E. w { x | A ( F |` B ) x } = { w } )
18	16, 17	sylibr 133	. . . 4 |- ( z e. ( ( F |` B ) ` A ) -> E! x A ( F |` B ) x )
19		euex 2030	. . . 4 |- ( E! x A ( F |` B ) x -> E. x A ( F |` B ) x )
20		df-br 3938	. . . . . . . 8 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F |` B ) )
21		df-res 4559	. . . . . . . . 9 |- ( F |` B ) = ( F i^i ( B X. _V ) )
22	21	eleq2i 2207	. . . . . . . 8 |- ( <. A , x >. e. ( F |` B ) <-> <. A , x >. e. ( F i^i ( B X. _V ) ) )
23	20, 22	bitri 183	. . . . . . 7 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F i^i ( B X. _V ) ) )
24		elin 3264	. . . . . . . 8 |- ( <. A , x >. e. ( F i^i ( B X. _V ) ) <-> ( <. A , x >. e. F /\ <. A , x >. e. ( B X. _V ) ) )
25	24	simprbi 273	. . . . . . 7 |- ( <. A , x >. e. ( F i^i ( B X. _V ) ) -> <. A , x >. e. ( B X. _V ) )
26	23, 25	sylbi 120	. . . . . 6 |- ( A ( F |` B ) x -> <. A , x >. e. ( B X. _V ) )
27		opelxp1 4581	. . . . . 6 |- ( <. A , x >. e. ( B X. _V ) -> A e. B )
28	26, 27	syl 14	. . . . 5 |- ( A ( F |` B ) x -> A e. B )
29	28	exlimiv 1578	. . . 4 |- ( E. x A ( F |` B ) x -> A e. B )
30	18, 19, 29	3syl 17	. . 3 |- ( z e. ( ( F |` B ) ` A ) -> A e. B )
31	30	con3i 622	. 2 |- ( -. A e. B -> -. z e. ( ( F |` B ) ` A ) )
32	31	eq0rdv 3412	1 |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   [wsb 1736   E!weu 2000   {cab 2126   _Vcvv 2689   i^i cin 3075   (/)c0 3368   {csn 3532   <.cop 3535   U.cuni 3744   class class class wbr 3937   X. cxp 4545   |` cres 4549   iotacio 5094   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-res 4559  df-iota 5096  df-fv 5139

This theorem is referenced by: (None)