Theorem fresin 5183

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 3220	. . 3 |- ( A i^i X ) C_ A
2		fssres 5180	. . 3 |- ( ( F : A --> B /\ ( A i^i X ) C_ A ) -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
3	1, 2	mpan2 416	. 2 |- ( F : A --> B -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
4		resres 4721	. . . 4 |- ( ( F |` A ) |` X ) = ( F |` ( A i^i X ) )
5		ffn 5155	. . . . . 6 |- ( F : A --> B -> F Fn A )
6		fnresdm 5117	. . . . . 6 |- ( F Fn A -> ( F |` A ) = F )
7	5, 6	syl 14	. . . . 5 |- ( F : A --> B -> ( F |` A ) = F )
8	7	reseq1d 4708	. . . 4 |- ( F : A --> B -> ( ( F |` A ) |` X ) = ( F |` X ) )
9	4, 8	syl5eqr 2134	. . 3 |- ( F : A --> B -> ( F |` ( A i^i X ) ) = ( F |` X ) )
10	9	feq1d 5143	. 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 145	1 |- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B )

Syntax hints:   -> wi 4   = wceq 1289   i^i cin 2998   C_ wss 2999   |` cres 4438   Fn wfn 5005   -->wf 5006

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-fun 5012  df-fn 5013  df-f 5014

This theorem is referenced by: (None)