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Theorem fvclss 7233
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 2731 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
2 tz6.12i 6910 . . . . . . . . . 10 (𝑦 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
31, 2biimtrid 241 . . . . . . . . 9 (𝑦 ≠ ∅ → (𝑦 = (𝐹𝑥) → 𝑥𝐹𝑦))
43eximdv 1912 . . . . . . . 8 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → ∃𝑥 𝑥𝐹𝑦))
5 vex 3470 . . . . . . . . 9 𝑦 ∈ V
65elrn 5884 . . . . . . . 8 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
74, 6imbitrrdi 251 . . . . . . 7 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → 𝑦 ∈ ran 𝐹))
87com12 32 . . . . . 6 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹))
98necon1bd 2950 . . . . 5 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 = ∅))
10 velsn 4637 . . . . 5 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10imbitrrdi 251 . . . 4 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1211orrd 860 . . 3 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1312ss2abi 4056 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
14 df-un 3946 . 2 (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
1513, 14sseqtrri 4012 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wne 2932  cun 3939  wss 3941  c0 4315  {csn 4621   class class class wbr 5139  ran crn 5668  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-iota 6486  df-fv 6542
This theorem is referenced by:  fvclex  7939
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