Theorem nfvres 5726

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 5380	. . . . . . . . . 10 |- ( ( F |` B ) ` A ) = ( iota x A ( F |` B ) x )
2		df-iota 5332	. . . . . . . . . 10 |- ( iota x A ( F |` B ) x ) = U. { y | { x | A ( F |` B ) x } = { y } }
3	1, 2	eqtri 2259	. . . . . . . . 9 |- ( ( F |` B ) ` A ) = U. { y | { x | A ( F |` B ) x } = { y } }
4	3	eleq2i 2305	. . . . . . . 8 |- ( z e. ( ( F |` B ) ` A ) <-> z e. U. { y | { x | A ( F |` B ) x } = { y } } )
5		eluni 3933	. . . . . . . 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 1671	. . . . . . 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 2225	. . . . . . . 8 |- ( w e. { y | { x | A ( F |` B ) x } = { y } } <-> [ w / y ] { x | A ( F |` B ) x } = { y } )
10		nfv 1581	. . . . . . . . 9 |- F/ y { x | A ( F |` B ) x } = { w }
11		sneq 3716	. . . . . . . . . 10 |- ( y = w -> { y } = { w } )
12	11	eqeq2d 2250	. . . . . . . . 9 |- ( y = w -> ( { x | A ( F |` B ) x } = { y } <-> { x | A ( F |` B ) x } = { w } ) )
13	10, 12	sbie 1844	. . . . . . . 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 1658	. . . . . 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 3776	. . . . 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 2116	. . . 4 |- ( E! x A ( F |` B ) x -> E. x A ( F |` B ) x )
20		df-br 4126	. . . . . . . 8 |- ( A ( F |` B ) x <-> <. A , x >. e. ( F |` B ) )
21		df-res 4781	. . . . . . . . 9 |- ( F |` B ) = ( F i^i ( B X. _V ) )
22	21	eleq2i 2305	. . . . . . . 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 3412	. . . . . . . 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 4803	. . . . . 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 1651	. . . 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 641	. 2 |- ( -. A e. B -> -. z e. ( ( F |` B ) ` A ) )
32	31	eq0rdv 3570	1 |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   [wsb 1815   E!weu 2086   e. wcel 2209   {cab 2224   _Vcvv 2821   i^i cin 3219   (/)c0 3520   {csn 3705   <.cop 3708   U.cuni 3930   class class class wbr 4125   X. cxp 4767   |` cres 4771   iotacio 5330   ` cfv 5372

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-res 4781  df-iota 5332  df-fv 5380

This theorem is referenced by: (None)