Theorem staffn 20024

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 20022	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
5		dffn5 6810	. . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
6	5	biimpi 215	. 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
7	4, 6	eqtr4id 2798	1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Syntax hints:   → wi 4   = wceq 1539   ↦ cmpt 5153   Fn wfn 6413  ‘cfv 6418  Basecbs 16840  *_𝑟cstv 16890  *_rfcstf 20018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-staf 20020

This theorem is referenced by: (None)