ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  strfvssn GIF version

Theorem strfvssn 12454
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 12452 . 2 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
53elexd 2750 . . 3 (𝜑𝑁 ∈ V)
6 fvssunirng 5525 . . 3 (𝑁 ∈ V → (𝑆𝑁) ⊆ ran 𝑆)
75, 6syl 14 . 2 (𝜑 → (𝑆𝑁) ⊆ ran 𝑆)
84, 7eqsstrd 3191 1 (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129   cuni 3807  ran crn 4623  cfv 5211  cn 8895  Slot cslot 12431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fv 5219  df-slot 12436
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator