Theorem fresin 5296

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 3291	. . 3 |- ( A i^i X ) C_ A
2		fssres 5293	. . 3 |- ( ( F : A --> B /\ ( A i^i X ) C_ A ) -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
3	1, 2	mpan2 421	. 2 |- ( F : A --> B -> ( F |` ( A i^i X ) ) : ( A i^i X ) --> B )
4		resres 4826	. . . 4 |- ( ( F |` A ) |` X ) = ( F |` ( A i^i X ) )
5		ffn 5267	. . . . . 6 |- ( F : A --> B -> F Fn A )
6		fnresdm 5227	. . . . . 6 |- ( F Fn A -> ( F |` A ) = F )
7	5, 6	syl 14	. . . . 5 |- ( F : A --> B -> ( F |` A ) = F )
8	7	reseq1d 4813	. . . 4 |- ( F : A --> B -> ( ( F |` A ) |` X ) = ( F |` X ) )
9	4, 8	syl5eqr 2184	. . 3 |- ( F : A --> B -> ( F |` ( A i^i X ) ) = ( F |` X ) )
10	9	feq1d 5254	. 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 1331   i^i cin 3065   C_ wss 3066   |` cres 4536   Fn wfn 5113   -->wf 5114

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-fun 5120  df-fn 5121  df-f 5122

This theorem is referenced by:  limcresi  12793