Theorem fvrn0 6860

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 4120	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 5242	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4607	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3919	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	eqeltrdi 2845	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 4119	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6824	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 147	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvexd 6847	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
11		fvbr0 6859	. . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
12	11	ori 862	. . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
13	12	con1i 147	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋))
14		brelrng 5888	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹)
15	9, 10, 13, 14	syl3anc 1374	. . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
16	7, 15	sselid 3920	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
17	6, 16	pm2.61i 182	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888  ∅c0 4274  {csn 4568   class class class wbr 5086  ran crn 5623  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5630  df-dm 5632  df-rn 5633  df-iota 6446  df-fv 6498

This theorem is referenced by:  fvn0fvelrn  6861  orderseqlem  8098  dfac4  10033  dfac2b  10042  dfacacn  10053  axdc2lem  10359  axcclem  10368  seqexw  13941  plusffval  18572  grpsubfval  18917  mulgfval  19003  staffval  20776  scaffval  20833  lpival  21281  ipffval  21605  nmfval  24531  tcphex  25162  tchnmfval  25173  rrnval  38139  lsatset  39427  fvnonrel  44027