Theorem strnfvn 12968

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12953) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12992. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
strnfvn	⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Proof of Theorem strnfvn

Step	Hyp	Ref	Expression
1		strnfvn.c	. . 3 ⊢ 𝐸 = Slot 𝑁
2		strnfvn.f	. . . 4 ⊢ 𝑆 ∈ V
3	2	a1i 9	. . 3 ⊢ (⊤ → 𝑆 ∈ V)
4		strnfvn.n	. . . 4 ⊢ 𝑁 ∈ ℕ
5	4	a1i 9	. . 3 ⊢ (⊤ → 𝑁 ∈ ℕ)
6	1, 3, 5	strnfvnd 12967	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	mptru 1382	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1373  ⊤wtru 1374   ∈ wcel 2178  Vcvv 2776  ‘cfv 5290  ℕcn 9071  Slot cslot 12946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-slot 12951

This theorem is referenced by:  ndxarg  12970  strsl0  12996  baseval  13000