Theorem fresin 5088

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 3186	. . 3 |- ( A i^i X ) C_ A
2		fssres 5086	. . 3 |- ( ( F : A --> B /\ ( A i^i X ) C_ A ) -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
3	1, 2	mpan2 415	. 2 |- ( F : A --> B -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
4		resres 4642	. . . 4 |- ( ( F |` A ) |` X ) = ( F |` ( A i^i X ) )
5		ffn 5066	. . . . . 6 |- ( F : A --> B -> F Fn A )
6		fnresdm 5028	. . . . . 6 |- ( F Fn A -> ( F |` A ) = F )
7	5, 6	syl 14	. . . . 5 |- ( F : A --> B -> ( F |` A ) = F )
8	7	reseq1d 4629	. . . 4 |- ( F : A --> B -> ( ( F |` A ) |` X ) = ( F |` X ) )
9	4, 8	syl5eqr 2127	. . 3 |- ( F : A --> B -> ( F |` ( A i^i X ) ) = ( F |` X ) )
10	9	feq1d 5054	. 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 1284   i^i cin 2972   C_ wss 2973   |` cres 4365   Fn wfn 4917   -->wf 4918

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-fun 4924  df-fn 4925  df-f 4926

This theorem is referenced by: (None)