Theorem respreima 5597

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 5202	. . 3 |- ( Fun F <-> F Fn dom F )
2		elin 3291	. . . . . . . . 9 |- ( x e. ( B i^i dom F ) <-> ( x e. B /\ x e. dom F ) )
3		ancom 264	. . . . . . . . 9 |- ( ( x e. B /\ x e. dom F ) <-> ( x e. dom F /\ x e. B ) )
4	2, 3	bitri 183	. . . . . . . 8 |- ( x e. ( B i^i dom F ) <-> ( x e. dom F /\ x e. B ) )
5	4	anbi1i 454	. . . . . . 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 5494	. . . . . . . . . 10 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7	6	eleq1d 2226	. . . . . . . . 9 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
8	7	adantl 275	. . . . . . . 8 |- ( ( x e. dom F /\ x e. B ) -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
9	8	pm5.32i 450	. . . . . . 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 183	. . . . . 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 552	. . . . 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 195	. . . 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 5269	. . . . . . . 8 |- ( F Fn dom F -> Fun F )
15		funres 5213	. . . . . . . 8 |- ( Fun F -> Fun ( F |` B ) )
16	14, 15	syl 14	. . . . . . 7 |- ( F Fn dom F -> Fun ( F |` B ) )
17		dmres 4889	. . . . . . 7 |- dom ( F |` B ) = ( B i^i dom F )
18	16, 17	jctir 311	. . . . . 6 |- ( F Fn dom F -> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
19		df-fn 5175	. . . . . 6 |- ( ( F |` B ) Fn ( B i^i dom F ) <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
20	18, 19	sylibr 133	. . . . 5 |- ( F Fn dom F -> ( F |` B ) Fn ( B i^i dom F ) )
21		elpreima 5588	. . . . 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 3291	. . . . 5 |- ( x e. ( ( `' F " A ) i^i B ) <-> ( x e. ( `' F " A ) /\ x e. B ) )
24		elpreima 5588	. . . . . 6 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
25	24	anbi1d 461	. . . . 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	syl5bb 191	. . . 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 219	. . 3 |- ( F Fn dom F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
28	1, 27	sylbi 120	. 2 |- ( Fun F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
29	28	eqrdv 2155	1 |- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   i^i cin 3101   `'ccnv 4587   dom cdm 4588   |` cres 4590   "cima 4591   Fun wfun 5166   Fn wfn 5167   ` cfv 5172

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180

This theorem is referenced by: (None)