Theorem fvrn0 6870

Description: A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)

Assertion

Ref	Expression
fvrn0	⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Proof of Theorem fvrn0

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) = ∅)
2		ssun2 4138	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 5257	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4622	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3940	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	eqeltrdi 2836	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 4137	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6832	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 147	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvexd 6855	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
11		fvbr0 6869	. . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
12	11	ori 861	. . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
13	12	con1i 147	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋))
14		brelrng 5894	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹)
15	9, 10, 13, 14	syl3anc 1373	. . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
16	7, 15	sselid 3941	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
17	6, 16	pm2.61i 182	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909  ∅c0 4292  {csn 4585   class class class wbr 5102  ran crn 5632  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6452  df-fv 6507

This theorem is referenced by:  fvn0fvelrn  6871  fvssunirnOLD  6874  orderseqlem  8113  dfac4  10051  dfac2b  10060  dfacacn  10071  axdc2lem  10377  axcclem  10386  seqexw  13958  plusffval  18549  grpsubfval  18891  mulgfval  18977  staffval  20726  scaffval  20762  lpival  21210  ipffval  21533  nmfval  24452  tcphex  25093  tchnmfval  25104  rrnval  37794  lsatset  38956  fvnonrel  43559