Theorem respreima 5624

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 5228	. . 3 |- ( Fun F <-> F Fn dom F )
2		elin 3310	. . . . . . . . 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 455	. . . . . . 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 5520	. . . . . . . . . 10 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7	6	eleq1d 2239	. . . . . . . . 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 451	. . . . . . 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 557	. . . . 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 5295	. . . . . . . 8 |- ( F Fn dom F -> Fun F )
15		funres 5239	. . . . . . . 8 |- ( Fun F -> Fun ( F |` B ) )
16	14, 15	syl 14	. . . . . . 7 |- ( F Fn dom F -> Fun ( F |` B ) )
17		dmres 4912	. . . . . . 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 5201	. . . . . 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 5615	. . . . 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 3310	. . . . 5 |- ( x e. ( ( `' F " A ) i^i B ) <-> ( x e. ( `' F " A ) /\ x e. B ) )
24		elpreima 5615	. . . . . 6 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
25	24	anbi1d 462	. . . . 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 2168	1 |- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   i^i cin 3120   `'ccnv 4610   dom cdm 4611   |` cres 4613   "cima 4614   Fun wfun 5192   Fn wfn 5193   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)