MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  strfvss Structured version   Visualization version   GIF version

Theorem strfvss 15797
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 15794 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
4 fvssunirn 6175 . . 3 (𝑆𝑁) ⊆ ran 𝑆
53, 4syl6eqss 3639 . 2 (𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
6 fvprc 6144 . . 3 𝑆 ∈ V → (𝐸𝑆) = ∅)
7 0ss 3949 . . 3 ∅ ⊆ ran 𝑆
86, 7syl6eqss 3639 . 2 𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
95, 8pm2.61i 176 1 (𝐸𝑆) ⊆ ran 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1992  Vcvv 3191  wss 3560  c0 3896   cuni 4407  ran crn 5080  cfv 5850  Slot cslot 15775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858  df-slot 15780
This theorem is referenced by:  wunstr  15798  prdsval  16031  prdsbas  16033  prdsplusg  16034  prdsmulr  16035  prdsvsca  16036  prdshom  16043
  Copyright terms: Public domain W3C validator