Theorem fvclss 2907

Description: Upper bound for the class of values of a class.

Assertion

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

Distinct variable group(s):   x,y,F

Proof of Theorem fvclss

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
2	1	tz6.12i 2847	. . . . . . . . . 10 ⊢ (¬ y = ∅ → ((F ‘x) = y → xFy))
3		cleqcom 1103	. . . . . . . . . 10 ⊢ (y = (F ‘x) ↔ (F ‘x) = y)
4	2, 3	syl5ib 181	. . . . . . . . 9 ⊢ (¬ y = ∅ → (y = (F ‘x) → xFy))
5	4	19.22dv 947	. . . . . . . 8 ⊢ (¬ y = ∅ → (∃x y = (F ‘x) → ∃x xFy))
6		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
7	6	elrn2 2563	. . . . . . . 8 ⊢ (y ∈ ran F ↔ ∃x xFy)
8	5, 7	syl6ibr 186	. . . . . . 7 ⊢ (¬ y = ∅ → (∃x y = (F ‘x) → y ∈ ran F))
9	8	com12 13	. . . . . 6 ⊢ (∃x y = (F ‘x) → (¬ y = ∅ → y ∈ ran F))
10	9	con1d 85	. . . . 5 ⊢ (∃x y = (F ‘x) → (¬ y ∈ ran F → y = ∅))
11		elsn 1820	. . . . 5 ⊢ (y ∈ {∅} ↔ y = ∅)
12	10, 11	syl6ibr 186	. . . 4 ⊢ (∃x y = (F ‘x) → (¬ y ∈ ran F → y ∈ {∅}))
13	12	orrd 203	. . 3 ⊢ (∃x y = (F ‘x) → (y ∈ ran F ∨ y ∈ {∅}))
14	13	ss2abi 1552	. 2 ⊢ {y∣∃x y = (F ‘x)} ⊆ {y∣(y ∈ ran F ∨ y ∈ {∅})}
15		df-un 1490	. 2 ⊢ (ran F ∪ {∅}) = {y∣(y ∈ ran F ∨ y ∈ {∅})}
16	14, 15	sseqtr4 1533	1 ⊢ {y∣∃x y = (F ‘x)} ⊆ (ran F ∪ {∅})

Syntax hints:  ¬ wn 1   ∨ wo 195  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054  ran crn 2411   ‘cfv 2422

This theorem is referenced by:  fvclex 2908

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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