Theorem staffn 17816

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 5893	. . 3 |- ( .* Fn B <-> .* = ( x e. B |-> ( .* ` x ) ) )
2	1	biimpi 194	. 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 17814	. 2 |- .xb = ( x e. B |-> ( .* ` x ) )
7	2, 6	syl6reqr 2462	1 |- ( .* Fn B -> .xb = .* )

Syntax hints:   -> wi 4   = wceq 1405   |-> cmpt 4452   Fn wfn 5563   ` cfv 5568   Basecbs 14839   *rcstv 14909   *rfcstf 17810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-staf 17812

This theorem is referenced by: (None)