Theorem nfvres 5519

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 5196	. . . . . . . . . 10 |- ( ( F |` B ) ` A ) = ( iota x A ( F |` B ) x )
2		df-iota 5153	. . . . . . . . . 10 |- ( iota x A ( F |` B ) x ) = U. { y | { x | A ( F |` B ) x } = { y } }
3	1, 2	eqtri 2186	. . . . . . . . 9 |- ( ( F |` B ) ` A ) = U. { y | { x | A ( F |` B ) x } = { y } }
4	3	eleq2i 2233	. . . . . . . 8 |- ( z e. ( ( F |` B ) ` A ) <-> z e. U. { y | { x | A ( F |` B ) x } = { y } } )
5		eluni 3792	. . . . . . . 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 1606	. . . . . . 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 2152	. . . . . . . 8 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> [ w / y ] { x | A ( F |` B ) x } = { y } )
10		nfv 1516	. . . . . . . . 9 |- F/ y { x | A ( F |` B ) x } = { w }
11		sneq 3587	. . . . . . . . . 10 |- ( y = w -> { y } = { w } )
12	11	eqeq2d 2177	. . . . . . . . 9 |- ( y = w -> ( { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } ) )
13	10, 12	sbie 1779	. . . . . . . 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 1593	. . . . . 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 3645	. . . . 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 2044	. . . 4 |- ( E! x A ( F |` B ) x -> E. x A ( F |` B ) x )
20		df-br 3983	. . . . . . . 8 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F |` B ) )
21		df-res 4616	. . . . . . . . 9 |- ( F |` B ) = ( F i^i ( B X. _V ) )
22	21	eleq2i 2233	. . . . . . . 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 3305	. . . . . . . 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 4638	. . . . . 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 1586	. . . 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 3453	1 |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   [wsb 1750   E!weu 2014   e. wcel 2136   {cab 2151   _Vcvv 2726   i^i cin 3115   (/)c0 3409   {csn 3576   <.cop 3579   U.cuni 3789   class class class wbr 3982   X. cxp 4602   |` cres 4606   iotacio 5151   ` cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-res 4616  df-iota 5153  df-fv 5196

This theorem is referenced by: (None)