Theorem stafval 20727

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 6840	. 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 20726	. 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
6		fvex 6853	. 2 ⊢ ( ∗ ‘𝐴) ∈ V
7	1, 5, 6	fvmpt 6950	1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  Basecbs 17155  *_𝑟cstv 17198  *_rfcstf 20722

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-staf 20724

This theorem is referenced by:  srngcl  20734  srngnvl  20735  srngadd  20736  srngmul  20737  srng1  20738  srng0  20739  issrngd  20740  iporthcom  21520