Theorem staffn 17049

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 5839	. . 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 17047	. 2 |- .xb = ( x e. B |-> ( .* ` x ) )
7	2, 6	syl6reqr 2511	1 |- ( .* Fn B -> .xb = .* )

Syntax hints:   -> wi 4   = wceq 1370   |-> cmpt 4451   Fn wfn 5514   ` cfv 5519   Basecbs 14285   *rcstv 14351   *rfcstf 17043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-staf 17045

This theorem is referenced by: (None)