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Theorem strfvssn 11995
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.) (Revised by Jim Kingdon, 19-Jan-2023.)
Hypotheses
Ref Expression
strfvssn.c 𝐸 = Slot 𝑁
strfvssn.s (𝜑𝑆𝑉)
strfvssn.n (𝜑𝑁 ∈ ℕ)
Assertion
Ref Expression
strfvssn (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)

Proof of Theorem strfvssn
StepHypRef Expression
1 strfvssn.c . . 3 𝐸 = Slot 𝑁
2 strfvssn.s . . 3 (𝜑𝑆𝑉)
3 strfvssn.n . . 3 (𝜑𝑁 ∈ ℕ)
41, 2, 3strnfvnd 11993 . 2 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
53elexd 2699 . . 3 (𝜑𝑁 ∈ V)
6 fvssunirng 5436 . . 3 (𝑁 ∈ V → (𝑆𝑁) ⊆ ran 𝑆)
75, 6syl 14 . 2 (𝜑 → (𝑆𝑁) ⊆ ran 𝑆)
84, 7eqsstrd 3133 1 (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071   cuni 3736  ran crn 4540  cfv 5123  cn 8732  Slot cslot 11972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-slot 11977
This theorem is referenced by: (None)
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