Theorem respreima 5646

Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
respreima	|- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Proof of Theorem respreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 5248	. . 3 |- ( Fun F <-> F Fn dom F )
2		elin 3320	. . . . . . . . 9 |- ( x e. ( B i^i dom F ) <-> ( x e. B /\ x e. dom F ) )
3		ancom 266	. . . . . . . . 9 |- ( ( x e. B /\ x e. dom F ) <-> ( x e. dom F /\ x e. B ) )
4	2, 3	bitri 184	. . . . . . . 8 |- ( x e. ( B i^i dom F ) <-> ( x e. dom F /\ x e. B ) )
5	4	anbi1i 458	. . . . . . 7 |- ( ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) <-> ( ( x e. dom F /\ x e. B ) /\ ( ( F |` B ) ` x ) e. A ) )
6		fvres 5541	. . . . . . . . . 10 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7	6	eleq1d 2246	. . . . . . . . 9 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
8	7	adantl 277	. . . . . . . 8 |- ( ( x e. dom F /\ x e. B ) -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
9	8	pm5.32i 454	. . . . . . 7 |- ( ( ( x e. dom F /\ x e. B ) /\ ( ( F |` B ) ` x ) e. A ) <-> ( ( x e. dom F /\ x e. B ) /\ ( F ` x ) e. A ) )
10	5, 9	bitri 184	. . . . . 6 |- ( ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) <-> ( ( x e. dom F /\ x e. B ) /\ ( F ` x ) e. A ) )
11	10	a1i 9	. . . . 5 |- ( F Fn dom F -> ( ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) <-> ( ( x e. dom F /\ x e. B ) /\ ( F ` x ) e. A ) ) )
12		an32 562	. . . . 5 |- ( ( ( x e. dom F /\ x e. B ) /\ ( F ` x ) e. A ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) /\ x e. B ) )
13	11, 12	bitrdi 196	. . . 4 |- ( F Fn dom F -> ( ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) /\ x e. B ) ) )
14		fnfun 5315	. . . . . . . 8 |- ( F Fn dom F -> Fun F )
15		funres 5259	. . . . . . . 8 |- ( Fun F -> Fun ( F |` B ) )
16	14, 15	syl 14	. . . . . . 7 |- ( F Fn dom F -> Fun ( F |` B ) )
17		dmres 4930	. . . . . . 7 |- dom ( F |` B ) = ( B i^i dom F )
18	16, 17	jctir 313	. . . . . 6 |- ( F Fn dom F -> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
19		df-fn 5221	. . . . . 6 |- ( ( F |` B ) Fn ( B i^i dom F ) <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
20	18, 19	sylibr 134	. . . . 5 |- ( F Fn dom F -> ( F |` B ) Fn ( B i^i dom F ) )
21		elpreima 5637	. . . . 5 |- ( ( F |` B ) Fn ( B i^i dom F ) -> ( x e. ( `' ( F |` B ) " A ) <-> ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) ) )
22	20, 21	syl 14	. . . 4 |- ( F Fn dom F -> ( x e. ( `' ( F |` B ) " A ) <-> ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) ) )
23		elin 3320	. . . . 5 |- ( x e. ( ( `' F " A ) i^i B ) <-> ( x e. ( `' F " A ) /\ x e. B ) )
24		elpreima 5637	. . . . . 6 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
25	24	anbi1d 465	. . . . 5 |- ( F Fn dom F -> ( ( x e. ( `' F " A ) /\ x e. B ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) /\ x e. B ) ) )
26	23, 25	bitrid 192	. . . 4 |- ( F Fn dom F -> ( x e. ( ( `' F " A ) i^i B ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) /\ x e. B ) ) )
27	13, 22, 26	3bitr4d 220	. . 3 |- ( F Fn dom F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
28	1, 27	sylbi 121	. 2 |- ( Fun F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
29	28	eqrdv 2175	1 |- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   i^i cin 3130   `'ccnv 4627   dom cdm 4628   |` cres 4630   "cima 4631   Fun wfun 5212   Fn wfn 5213   ` cfv 5218

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by: (None)