Theorem staffn 17281

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 5911	. . 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 17279	. 2 |- .xb = ( x e. B |-> ( .* ` x ) )
7	2, 6	syl6reqr 2527	1 |- ( .* Fn B -> .xb = .* )

Syntax hints:   -> wi 4   = wceq 1379   |-> cmpt 4505   Fn wfn 5581   ` cfv 5586   Basecbs 14486   *rcstv 14553   *rfcstf 17275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-staf 17277

This theorem is referenced by: (None)