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Theorem fvclss 6643
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 2778 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
2 tz6.12i 6355 . . . . . . . . . 10 (𝑦 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
31, 2syl5bi 232 . . . . . . . . 9 (𝑦 ≠ ∅ → (𝑦 = (𝐹𝑥) → 𝑥𝐹𝑦))
43eximdv 1998 . . . . . . . 8 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → ∃𝑥 𝑥𝐹𝑦))
5 vex 3354 . . . . . . . . 9 𝑦 ∈ V
65elrn 5504 . . . . . . . 8 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
74, 6syl6ibr 242 . . . . . . 7 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → 𝑦 ∈ ran 𝐹))
87com12 32 . . . . . 6 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹))
98necon1bd 2961 . . . . 5 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 = ∅))
10 velsn 4332 . . . . 5 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10syl6ibr 242 . . . 4 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1211orrd 850 . . 3 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1312ss2abi 3823 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
14 df-un 3728 . 2 (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
1513, 14sseqtr4i 3787 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 834   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  cun 3721  wss 3723  c0 4063  {csn 4316   class class class wbr 4786  ran crn 5250  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-iota 5994  df-fv 6039
This theorem is referenced by:  fvclex  7285
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