Theorem staffn 18849

Description: The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)

Hypotheses

Ref	Expression
staffval.b	|- B = ( Base ` R )
staffval.i	|- .* = ( *r ` R )
staffval.f	|- .xb = ( *rf ` R )

Assertion

Ref	Expression
staffn	|- ( .* Fn B -> .xb = .* )

Proof of Theorem staffn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6241	. . 3 |- ( .* Fn B <-> .* = ( x e. B |-> ( .* ` x ) ) )
2	1	biimpi 206	. 2 |- ( .* Fn B -> .* = ( x e. B |-> ( .* ` x ) ) )
3		staffval.b	. . 3 |- B = ( Base ` R )
4		staffval.i	. . 3 |- .* = ( *r ` R )
5		staffval.f	. . 3 |- .xb = ( *rf ` R )
6	3, 4, 5	staffval 18847	. 2 |- .xb = ( x e. B |-> ( .* ` x ) )
7	2, 6	syl6reqr 2675	1 |- ( .* Fn B -> .xb = .* )

Syntax hints:   -> wi 4   = wceq 1483   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888   Basecbs 15857   *rcstv 15943   *rfcstf 18843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-staf 18845

This theorem is referenced by: (None)