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Theorem strfvss 16087
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 16083 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
4 fvssunirn 6358 . . 3 (𝑆𝑁) ⊆ ran 𝑆
53, 4syl6eqss 3804 . 2 (𝑆 ∈ V → (𝐸𝑆) ⊆ ran 𝑆)
6 fvprc 6326 . . 3 𝑆 ∈ V → (𝐸𝑆) = ∅)
7 0ss 4116 . . 3 ∅ ⊆ ran 𝑆
86, 7syl6eqss 3804 . 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 1631  wcel 2145  Vcvv 3351  wss 3723  c0 4063   cuni 4574  ran crn 5250  cfv 6031  Slot cslot 16063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16068
This theorem is referenced by:  wunstr  16088  prdsval  16323  prdsbas  16325  prdsplusg  16326  prdsmulr  16327  prdsvsca  16328  prdshom  16335
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