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Theorem fvclss 7251
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 2735 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
2 tz6.12i 6925 . . . . . . . . . 10 (𝑦 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
31, 2biimtrid 241 . . . . . . . . 9 (𝑦 ≠ ∅ → (𝑦 = (𝐹𝑥) → 𝑥𝐹𝑦))
43eximdv 1913 . . . . . . . 8 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → ∃𝑥 𝑥𝐹𝑦))
5 vex 3475 . . . . . . . . 9 𝑦 ∈ V
65elrn 5896 . . . . . . . 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 2955 . . . . 5 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 = ∅))
10 velsn 4645 . . . . 5 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10imbitrrdi 251 . . . 4 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1211orrd 862 . . 3 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1312ss2abi 4061 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
14 df-un 3952 . 2 (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
1513, 14sseqtrri 4017 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wne 2937  cun 3945  wss 3947  c0 4323  {csn 4629   class class class wbr 5148  ran crn 5679  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5686  df-dm 5688  df-rn 5689  df-iota 6500  df-fv 6556
This theorem is referenced by:  fvclex  7962
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