Theorem stafval 21011

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 6879	. 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 21010	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
6		fvex 6892	. 2 ⊢ ( ∗ ‘𝐴) ∈ V
7	1, 5, 6	fvmpt 6987	1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  Basecbs 17304  *_𝑟cstv 17347  *_rfcstf 21006

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-staf 21008

This theorem is used by:  srngcl  21018  srngnvl  21019  srngadd  21020  srngmul  21021  srng1  21022  srng0  21023  issrngd  21024  iporthcom  21851