Theorem fresin 5068

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 3157	. . 3 |- ( A i^i X ) C_ A
2		fssres 5066	. . 3 |- ( ( F : A --> B /\ ( A i^i X ) C_ A ) -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
3	1, 2	mpan2 401	. 2 |- ( F : A --> B -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
4		resres 4624	. . . 4 |- ( ( F |` A ) |` X ) = ( F |` ( A i^i X ) )
5		ffn 5046	. . . . . 6 |- ( F : A --> B -> F Fn A )
6		fnresdm 5008	. . . . . 6 |- ( F Fn A -> ( F |` A ) = F )
7	5, 6	syl 14	. . . . 5 |- ( F : A --> B -> ( F |` A ) = F )
8	7	reseq1d 4611	. . . 4 |- ( F : A --> B -> ( ( F |` A ) |` X ) = ( F |` X ) )
9	4, 8	syl5eqr 2086	. . 3 |- ( F : A --> B -> ( F |` ( A i^i X ) ) = ( F |` X ) )
10	9	feq1d 5034	. 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 135	1 |- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B )

Syntax hints:   -> wi 4   = wceq 1243   i^i cin 2916   C_ wss 2917   |` cres 4347   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)