Theorem strnfvn 13222

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 13207) 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 13246. (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 13221	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	mptru 1407	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1398  ⊤wtru 1399   ∈ wcel 2203  Vcvv 2812  ‘cfv 5351  ℕcn 9233  Slot cslot 13200

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fv 5359  df-slot 13205

This theorem is referenced by:  ndxarg  13224  strsl0  13250  baseval  13254