Theorem respreima 5775

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 5356	. . 3 |- ( Fun F <-> F Fn dom F )
2		elin 3390	. . . . . . . . 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 5663	. . . . . . . . . 10 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7	6	eleq1d 2300	. . . . . . . . 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 564	. . . . 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 5427	. . . . . . . 8 |- ( F Fn dom F -> Fun F )
15		funres 5367	. . . . . . . 8 |- ( Fun F -> Fun ( F |` B ) )
16	14, 15	syl 14	. . . . . . 7 |- ( F Fn dom F -> Fun ( F |` B ) )
17		dmres 5034	. . . . . . 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 5329	. . . . . 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 5766	. . . . 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 3390	. . . . 5 |- ( x e. ( ( `' F " A ) i^i B ) <-> ( x e. ( `' F " A ) /\ x e. B ) )
24		elpreima 5766	. . . . . 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 2229	1 |- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   i^i cin 3199   `'ccnv 4724   dom cdm 4725   |` cres 4727   "cima 4728   Fun wfun 5320   Fn wfn 5321   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)