Theorem fresin 5366

Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Assertion

Ref	Expression
fresin	|- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B )

Proof of Theorem fresin

Step	Hyp	Ref	Expression
1		inss1 3342	. . 3 |- ( A i^i X ) C_ A
2		fssres 5363	. . 3 |- ( ( F : A --> B /\ ( A i^i X ) C_ A ) -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
3	1, 2	mpan2 422	. 2 |- ( F : A --> B -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
4		resres 4896	. . . 4 |- ( ( F |` A ) |` X ) = ( F |` ( A i^i X ) )
5		ffn 5337	. . . . . 6 |- ( F : A --> B -> F Fn A )
6		fnresdm 5297	. . . . . 6 |- ( F Fn A -> ( F |` A ) = F )
7	5, 6	syl 14	. . . . 5 |- ( F : A --> B -> ( F |` A ) = F )
8	7	reseq1d 4883	. . . 4 |- ( F : A --> B -> ( ( F |` A ) |` X ) = ( F |` X ) )
9	4, 8	eqtr3id 2213	. . 3 |- ( F : A --> B -> ( F |` ( A i^i X ) ) = ( F |` X ) )
10	9	feq1d 5324	. 2 |- ( F : A --> B -> ( ( F |` ( A i^i X ) ) : ( A i^i X ) --> B <-> ( F |` X ) : ( A i^i X ) --> B ) )
11	3, 10	mpbid 146	1 |- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B )

Syntax hints:   -> wi 4   = wceq 1343   i^i cin 3115   C_ wss 3116   |` cres 4606   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  limcresi  13275