Theorem strnfvn 12853

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12838) 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 12877. (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 12852	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	mptru 1382	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1373  ⊤wtru 1374   ∈ wcel 2176  Vcvv 2772  ‘cfv 5271  ℕcn 9036  Slot cslot 12831

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fv 5279  df-slot 12836

This theorem is referenced by:  ndxarg  12855  strsl0  12881  baseval  12885