Theorem strnfvn 13053

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 13038) 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 13077. (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 13052	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	mptru 1404	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  Vcvv 2799  ‘cfv 5318  ℕcn 9110  Slot cslot 13031

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-slot 13036

This theorem is referenced by:  ndxarg  13055  strsl0  13081  baseval  13085