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Theorem strfvss 17224
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 17222 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
4 fvssunirn 6939 . . 3 (𝑆𝑁) ⊆ ran 𝑆
53, 4eqsstrdi 4028 . 2 (𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
6 fvprc 6898 . . 3 𝑆 ∈ V → (𝐸𝑆) = ∅)
7 0ss 4400 . . 3 ∅ ⊆ ran 𝑆
86, 7eqsstrdi 4028 . 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 1540  wcel 2108  Vcvv 3480  wss 3951  c0 4333   cuni 4907  ran crn 5686  cfv 6561  Slot cslot 17218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-slot 17219
This theorem is referenced by:  wunstr  17225  prdsvallem  17499  prdsval  17500  prdsbas  17502  prdsplusg  17503  prdsmulr  17504  prdsvsca  17505  prdshom  17512
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