Theorem stafval 20923

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 6882	. 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 20922	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
6		fvex 6895	. 2 ⊢ ( ∗ ‘𝐴) ∈ V
7	1, 5, 6	fvmpt 6990	1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  Basecbs 17269  *_𝑟cstv 17312  *_rfcstf 20918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-staf 20920

This theorem is referenced by:  srngcl  20930  srngnvl  20931  srngadd  20932  srngmul  20933  srng1  20934  srng0  20935  issrngd  20936  iporthcom  21754