Theorem staffn 20600

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 20598	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
5		dffn5 6949	. . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
6	5	biimpi 215	. 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
7	4, 6	eqtr4id 2789	1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Syntax hints:   → wi 4   = wceq 1539   ↦ cmpt 5230   Fn wfn 6537  ‘cfv 6542  Basecbs 17148  *_𝑟cstv 17203  *_rfcstf 20594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-staf 20596

This theorem is referenced by: (None)