Theorem strfvss 16816

Description: A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypothesis

Ref	Expression
strfvss.e	⊢ 𝐸 = Slot 𝑁

Assertion

Ref	Expression
strfvss	⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆

Proof of Theorem strfvss

Step	Hyp	Ref	Expression
1		strfvss.e	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		id 22	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 ∈ V)
3	1, 2	strfvnd 16814	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁))
4		fvssunirn 6785	. . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆
5	3, 4	eqsstrdi 3971	. 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
6		fvprc 6748	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
7		0ss 4327	. . 3 ⊢ ∅ ⊆ ∪ ran 𝑆
8	6, 7	eqsstrdi 3971	. 2 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
9	5, 8	pm2.61i 182	1 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ran crn 5581  ‘cfv 6418  Slot cslot 16810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-slot 16811

This theorem is referenced by:  wunstr  16817  prdsvallem  17082  prdsval  17083  prdsbas  17085  prdsplusg  17086  prdsmulr  17087  prdsvsca  17088  prdshom  17095