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Theorem strfvss 17067
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 (𝐸𝑆) ⊆ ran 𝑆

Proof of Theorem strfvss
StepHypRef Expression
1 strfvss.e . . . 4 𝐸 = Slot 𝑁
2 id 22 . . . 4 (𝑆 ∈ V → 𝑆 ∈ V)
31, 2strfvnd 17065 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
4 fvssunirn 6879 . . 3 (𝑆𝑁) ⊆ ran 𝑆
53, 4eqsstrdi 4002 . 2 (𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
6 fvprc 6838 . . 3 𝑆 ∈ V → (𝐸𝑆) = ∅)
7 0ss 4360 . . 3 ∅ ⊆ ran 𝑆
86, 7eqsstrdi 4002 . 2 𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
95, 8pm2.61i 182 1 (𝐸𝑆) ⊆ ran 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3447  wss 3914  c0 4286   cuni 4869  ran crn 5638  cfv 6500  Slot cslot 17061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-slot 17062
This theorem is referenced by:  wunstr  17068  prdsvallem  17344  prdsval  17345  prdsbas  17347  prdsplusg  17348  prdsmulr  17349  prdsvsca  17350  prdshom  17357
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