Theorem strfvn 17147

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17171) and 𝑁 is the index of the component. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 18270) where 𝑆 ∈ Poset.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. Alternatively, use strfv 17164 instead of strfvn 17147. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
strfvn.f	⊢ 𝑆 ∈ V
strfvn.c	⊢ 𝐸 = Slot 𝑁

Assertion

Ref	Expression
strfvn	⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Proof of Theorem strfvn

Step	Hyp	Ref	Expression
1		strfvn.c	. . 3 ⊢ 𝐸 = Slot 𝑁
2		strfvn.f	. . . 4 ⊢ 𝑆 ∈ V
3	2	a1i 11	. . 3 ⊢ (⊤ → 𝑆 ∈ V)
4	1, 3	strfvnd 17146	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
5	4	mptru 1554	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  Vcvv 3431  ‘cfv 6485  Slot cslot 17142

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-slot 17143

This theorem is referenced by:  str0  17150  ndxarg  17157  setsnid  17169  baseval  17172  resvsca  33415