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Theorem fvclss 6382
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 2616 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
2 tz6.12i 6109 . . . . . . . . . 10 (𝑦 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
31, 2syl5bi 230 . . . . . . . . 9 (𝑦 ≠ ∅ → (𝑦 = (𝐹𝑥) → 𝑥𝐹𝑦))
43eximdv 1832 . . . . . . . 8 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → ∃𝑥 𝑥𝐹𝑦))
5 vex 3175 . . . . . . . . 9 𝑦 ∈ V
65elrn 5274 . . . . . . . 8 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
74, 6syl6ibr 240 . . . . . . 7 (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹𝑥) → 𝑦 ∈ ran 𝐹))
87com12 32 . . . . . 6 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹))
98necon1bd 2799 . . . . 5 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 = ∅))
10 velsn 4140 . . . . 5 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10syl6ibr 240 . . . 4 (∃𝑥 𝑦 = (𝐹𝑥) → (¬ 𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1211orrd 391 . . 3 (∃𝑥 𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹𝑦 ∈ {∅}))
1312ss2abi 3636 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
14 df-un 3544 . 2 (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹𝑦 ∈ {∅})}
1513, 14sseqtr4i 3600 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 381   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  cun 3537  wss 3539  c0 3873  {csn 4124   class class class wbr 4577  ran crn 5029  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-iota 5754  df-fv 5798
This theorem is referenced by:  fvclex  7008
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