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Theorem strfvss 15713
 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
ndxarg.1 𝐸 = Slot 𝑁
Assertion
Ref Expression
strfvss (𝐸𝑆) ⊆ ran 𝑆

Proof of Theorem strfvss
StepHypRef Expression
1 ndxarg.1 . . . 4 𝐸 = Slot 𝑁
2 id 22 . . . 4 (𝑆 ∈ V → 𝑆 ∈ V)
31, 2strfvnd 15710 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
4 fvssunirn 6127 . . 3 (𝑆𝑁) ⊆ ran 𝑆
53, 4syl6eqss 3618 . 2 (𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
6 fvprc 6097 . . 3 𝑆 ∈ V → (𝐸𝑆) = ∅)
7 0ss 3924 . . 3 ∅ ⊆ ran 𝑆
86, 7syl6eqss 3618 . 2 𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
95, 8pm2.61i 175 1 (𝐸𝑆) ⊆ ran 𝑆
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ran crn 5039  ‘cfv 5804  Slot cslot 15694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699 This theorem is referenced by:  wunstr  15714  prdsval  15938  prdsbas  15940  prdsplusg  15941  prdsmulr  15942  prdsvsca  15943  prdshom  15950
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