Theorem nfvres 5595

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 5267	. . . . . . . . . 10 |- ( ( F |` B ) ` A ) = ( iota x A ( F |` B ) x )
2		df-iota 5220	. . . . . . . . . 10 |- ( iota x A ( F |` B ) x ) = U. { y | { x | A ( F |` B ) x } = { y } }
3	1, 2	eqtri 2217	. . . . . . . . 9 |- ( ( F |` B ) ` A ) = U. { y | { x | A ( F |` B ) x } = { y } }
4	3	eleq2i 2263	. . . . . . . 8 |- ( z e. ( ( F |` B ) ` A ) <-> z e. U. { y | { x | A ( F |` B ) x } = { y } } )
5		eluni 3843	. . . . . . . 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 184	. . . . . . 7 |- ( z e. ( ( F |` B ) ` A ) <-> E. w ( z e. w /\ w e. { y | { x | A ( F |` B ) x } = { y } } ) )
7		exsimpr 1632	. . . . . . 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 121	. . . . . 6 |- ( z e. ( ( F |` B ) ` A ) -> E. w w e. { y | { x | A ( F |` B ) x } = { y } } )
9		df-clab 2183	. . . . . . . 8 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> [ w / y ] { x | A ( F |` B ) x } = { y } )
10		nfv 1542	. . . . . . . . 9 |- F/ y { x | A ( F |` B ) x } = { w }
11		sneq 3634	. . . . . . . . . 10 |- ( y = w -> { y } = { w } )
12	11	eqeq2d 2208	. . . . . . . . 9 |- ( y = w -> ( { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } ) )
13	10, 12	sbie 1805	. . . . . . . 8 |- ( [ w / y ] { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } )
14	9, 13	bitri 184	. . . . . . 7 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> { x | A ( F |` B ) x } = { w } )
15	14	exbii 1619	. . . . . 6 |- ( E. w w e. { y | { x | A ( F |` B ) x } = { y } } <-> E. w { x | A ( F |` B ) x } = { w } )
16	8, 15	sylib 122	. . . . 5 |- ( z e. ( ( F |` B ) ` A ) -> E. w { x | A ( F |` B ) x } = { w } )
17		euabsn2 3692	. . . . 5 |- ( E! x A ( F |` B ) x <-> E. w { x | A ( F |` B ) x } = { w } )
18	16, 17	sylibr 134	. . . 4 |- ( z e. ( ( F |` B ) ` A ) -> E! x A ( F |` B ) x )
19		euex 2075	. . . 4 |- ( E! x A ( F |` B ) x -> E. x A ( F |` B ) x )
20		df-br 4035	. . . . . . . 8 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F |` B ) )
21		df-res 4676	. . . . . . . . 9 |- ( F |` B ) = ( F i^i ( B X. _V ) )
22	21	eleq2i 2263	. . . . . . . 8 |- ( <. A , x >. e. ( F |` B ) <-> <. A , x >. e. ( F i^i ( B X. _V ) ) )
23	20, 22	bitri 184	. . . . . . 7 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F i^i ( B X. _V ) ) )
24		elin 3347	. . . . . . . 8 |- ( <. A , x >. e. ( F i^i ( B X. _V ) ) <-> ( <. A , x >. e. F /\ <. A , x >. e. ( B X. _V ) ) )
25	24	simprbi 275	. . . . . . 7 |- ( <. A , x >. e. ( F i^i ( B X. _V ) ) -> <. A , x >. e. ( B X. _V ) )
26	23, 25	sylbi 121	. . . . . 6 |- ( A ( F |` B ) x -> <. A , x >. e. ( B X. _V ) )
27		opelxp1 4698	. . . . . 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 1612	. . . 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 633	. 2 |- ( -. A e. B -> -. z e. ( ( F |` B ) ` A ) )
32	31	eq0rdv 3496	1 |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   [wsb 1776   E!weu 2045   e. wcel 2167   {cab 2182   _Vcvv 2763   i^i cin 3156   (/)c0 3451   {csn 3623   <.cop 3626   U.cuni 3840   class class class wbr 4034   X. cxp 4662   |` cres 4666   iotacio 5218   ` cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-res 4676  df-iota 5220  df-fv 5267

This theorem is referenced by: (None)