Theorem staffn 19739

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 19737	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
5		dffn5 6728	. . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
6	5	biimpi 219	. 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
7	4, 6	eqtr4id 2792	1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )

Syntax hints:   → wi 4   = wceq 1542   ↦ cmpt 5110   Fn wfn 6334  ‘cfv 6339  Basecbs 16586  *_𝑟cstv 16670  *_rfcstf 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-staf 19735

This theorem is referenced by: (None)