Theorem staffn 20758

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 20756	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
5		dffn5 6921	. . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
6	5	biimpi 216	. 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
7	4, 6	eqtr4id 2784	1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Syntax hints:   → wi 4   = wceq 1540   ↦ cmpt 5190   Fn wfn 6508  ‘cfv 6513  Basecbs 17185  *_𝑟cstv 17228  *_rfcstf 20752

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-staf 20754

This theorem is referenced by: (None)