Theorem fvrn0 6937

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 4189	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4667	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3992	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	eqeltrdi 2847	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 4188	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6899	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 147	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvexd 6922	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
11		fvbr0 6936	. . . . . 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 5955	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹)
15	9, 10, 13, 14	syl3anc 1370	. . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
16	7, 15	sselid 3993	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
17	6, 16	pm2.61i 182	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961  ∅c0 4339  {csn 4631   class class class wbr 5148  ran crn 5690  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571

This theorem is referenced by:  fvn0fvelrn  6938  fvssunirnOLD  6941  orderseqlem  8181  dfac4  10160  dfac2b  10169  dfacacn  10180  axdc2lem  10486  axcclem  10495  seqexw  14055  plusffval  18672  grpsubfval  19014  mulgfval  19100  staffval  20859  scaffval  20895  lpival  21352  ipffval  21684  nmfval  24617  tcphex  25265  tchnmfval  25276  rrnval  37814  lsatset  38972  fvnonrel  43587