Theorem respreima 6344

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 5918	. . 3 |- ( Fun F <-> F Fn dom F )
2		elin 3796	. . . . . . . . 9 |- ( x e. ( B i^i dom F ) <-> ( x e. B /\ x e. dom F ) )
3		ancom 466	. . . . . . . . 9 |- ( ( x e. B /\ x e. dom F ) <-> ( x e. dom F /\ x e. B ) )
4	2, 3	bitri 264	. . . . . . . 8 |- ( x e. ( B i^i dom F ) <-> ( x e. dom F /\ x e. B ) )
5	4	anbi1i 731	. . . . . . 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 6207	. . . . . . . . . 10 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7	6	eleq1d 2686	. . . . . . . . 9 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
8	7	adantl 482	. . . . . . . 8 |- ( ( x e. dom F /\ x e. B ) -> ( ( ( F |` B ) ` x ) e. A <-> ( F ` x ) e. A ) )
9	8	pm5.32i 669	. . . . . . 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 264	. . . . . 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 11	. . . . 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 839	. . . . 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	syl6bb 276	. . . 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 5988	. . . . . . . 8 |- ( F Fn dom F -> Fun F )
15		funres 5929	. . . . . . . 8 |- ( Fun F -> Fun ( F |` B ) )
16	14, 15	syl 17	. . . . . . 7 |- ( F Fn dom F -> Fun ( F |` B ) )
17		dmres 5419	. . . . . . 7 |- dom ( F |` B ) = ( B i^i dom F )
18	16, 17	jctir 561	. . . . . 6 |- ( F Fn dom F -> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
19		df-fn 5891	. . . . . 6 |- ( ( F |` B ) Fn ( B i^i dom F ) <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = ( B i^i dom F ) ) )
20	18, 19	sylibr 224	. . . . 5 |- ( F Fn dom F -> ( F |` B ) Fn ( B i^i dom F ) )
21		elpreima 6337	. . . . 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 17	. . . 4 |- ( F Fn dom F -> ( x e. ( `' ( F |` B ) " A ) <-> ( x e. ( B i^i dom F ) /\ ( ( F |` B ) ` x ) e. A ) ) )
23		elin 3796	. . . . 5 |- ( x e. ( ( `' F " A ) i^i B ) <-> ( x e. ( `' F " A ) /\ x e. B ) )
24		elpreima 6337	. . . . . 6 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
25	24	anbi1d 741	. . . . 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 272	. . . 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 300	. . 3 |- ( F Fn dom F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
28	1, 27	sylbi 207	. 2 |- ( Fun F -> ( x e. ( `' ( F |` B ) " A ) <-> x e. ( ( `' F " A ) i^i B ) ) )
29	28	eqrdv 2620	1 |- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  paste  21098  restmetu  22375  eulerpartlemt  30433  smfres  40997