Theorem stafval 20106

Description: The functionalization of the involution component of a structure. (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
stafval	⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))

Proof of Theorem stafval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6771	. 2 ⊢ (𝑥 = 𝐴 → ( ∗ ‘𝑥) = ( ∗ ‘𝐴))
2		staffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		staffval.i	. . 3 ⊢ ∗ = (*𝑟‘𝑅)
4		staffval.f	. . 3 ⊢ ∙ = (*rf‘𝑅)
5	2, 3, 4	staffval 20105	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
6		fvex 6784	. 2 ⊢ ( ∗ ‘𝐴) ∈ V
7	1, 5, 6	fvmpt 6872	1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2110  ‘cfv 6432  Basecbs 16910  *_𝑟cstv 16962  *_rfcstf 20101

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 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-staf 20103

This theorem is referenced by:  srngcl  20113  srngnvl  20114  srngadd  20115  srngmul  20116  srng1  20117  srng0  20118  issrngd  20119  iporthcom  20838