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Theorem strslfv2 12032
Description: A variation on strslfv 12033 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strfv2.s 𝑆 ∈ V
strfv2.f Fun 𝑆
strslfv2.e (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
strfv2.n ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
Assertion
Ref Expression
strslfv2 (𝐶𝑉𝐶 = (𝐸𝑆))

Proof of Theorem strslfv2
StepHypRef Expression
1 strslfv2.e . 2 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
2 strfv2.s . . 3 𝑆 ∈ V
32a1i 9 . 2 (𝐶𝑉𝑆 ∈ V)
4 strfv2.f . . 3 Fun 𝑆
54a1i 9 . 2 (𝐶𝑉 → Fun 𝑆)
6 strfv2.n . . 3 ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
76a1i 9 . 2 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
8 id 19 . 2 (𝐶𝑉𝐶𝑉)
91, 3, 5, 7, 8strslfv2d 12031 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2687  cop 3531  ccnv 4542  Fun wfun 5121  cfv 5127  cn 8740  ndxcnx 11986  Slot cslot 11988
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-iota 5092  df-fun 5129  df-fv 5135  df-slot 11993
This theorem is referenced by: (None)
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