Theorem fvclss 7097

Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)

Assertion

Ref	Expression
fvclss	⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})

Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem fvclss

Step	Hyp	Ref	Expression
1		eqcom 2745	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
2		tz6.12i 6782	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑥𝐹𝑦))
3	1, 2	syl5bi 241	. . . . . . . . 9 ⊢ (𝑦 ≠ ∅ → (𝑦 = (𝐹‘𝑥) → 𝑥𝐹𝑦))
4	3	eximdv 1921	. . . . . . . 8 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → ∃𝑥 𝑥𝐹𝑦))
5		vex 3426	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elrn 5791	. . . . . . . 8 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
7	4, 6	syl6ibr 251	. . . . . . 7 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ ran 𝐹))
8	7	com12 32	. . . . . 6 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹))
9	8	necon1bd 2960	. . . . 5 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 = ∅))
10		velsn 4574	. . . . 5 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	9, 10	syl6ibr 251	. . . 4 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 ∈ {∅}))
12	11	orrd 859	. . 3 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅}))
13	12	ss2abi 3996	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})}
14		df-un 3888	. 2 ⊢ (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})}
15	13, 14	sseqtrri 3954	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070  ran crn 5581  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591  df-iota 6376  df-fv 6426

This theorem is referenced by:  fvclex  7775