Theorem strfvn 16499

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16483) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 17550) where 𝑆 ∈ Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 16525. (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 16496	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
5	4	mptru 1540	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1533  ⊤wtru 1534   ∈ wcel 2110  Vcvv 3494  ‘cfv 6349  Slot cslot 16476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-slot 16481

This theorem is referenced by:  ndxarg  16502  str0  16529  setsnid  16533  baseval  16536  ressbas  16548  resvsca  30898