Theorem staffn 20915

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	⊢ ∙ = (*rf‘𝑅)

Assertion

Ref	Expression
staffn	⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Proof of Theorem staffn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		staffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		staffval.i	. . 3 ⊢ ∗ = (*𝑟‘𝑅)
3		staffval.f	. . 3 ⊢ ∙ = (*rf‘𝑅)
4	1, 2, 3	staffval 20913	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
5		dffn5 6929	. . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
6	5	biimpi 219	. 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
7	4, 6	eqtr4id 2819	1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Syntax hints:   → wi 4   = wceq 1563   ↦ cmpt 5186   Fn wfn 6520  ‘cfv 6525  Basecbs 17259  *_𝑟cstv 17302  *_rfcstf 20909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-staf 20911

This theorem is referenced by: (None)