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Theorem fvclss 7240
Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
Assertion
Ref Expression
fvclss {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem fvclss
StepHypRef Expression
1 eqcom 2739 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
2 tz6.12i 6919 . . . . . . . . . 10 (𝑦 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
31, 2biimtrid 241 . . . . . . . . 9 (𝑦 ≠ ∅ → (𝑦 = (𝐹𝑥) → 𝑥𝐹𝑦))
43eximdv 1920 . . . . . . . 8 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → ∃𝑥 𝑥𝐹𝑦))
5 vex 3478 . . . . . . . . 9 𝑦 ∈ V
65elrn 5893 . . . . . . . 8 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
74, 6syl6ibr 251 . . . . . . 7 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → 𝑦 ∈ ran 𝐹))
87com12 32 . . . . . 6 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹))
98necon1bd 2958 . . . . 5 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 = ∅))
10 velsn 4644 . . . . 5 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10syl6ibr 251 . . . 4 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1211orrd 861 . . 3 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1312ss2abi 4063 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
14 df-un 3953 . 2 (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
1513, 14sseqtrri 4019 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  cun 3946  wss 3948  c0 4322  {csn 4628   class class class wbr 5148  ran crn 5677  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551
This theorem is referenced by:  fvclex  7944
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