Theorem fvrn0 6909

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 23	. . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) = ∅)
2		ssun2 4132	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 5270	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4628	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3934	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	eqeltrdi 2871	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 4131	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6873	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 148	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvexd 6896	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
11		fvbr0 6908	. . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
12	11	ori 874	. . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
13	12	con1i 148	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋))
14		brelrng 5931	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹)
15	9, 10, 13, 14	syl3anc 1398	. . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
16	7, 15	sselid 3935	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
17	6, 16	pm2.61i 184	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903  ∅c0 4286  {csn 4589   class class class wbr 5109  ran crn 5662  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-cnv 5669  df-dm 5671  df-rn 5672  df-iota 6492  df-fv 6544

This theorem is referenced by:  fvn0fvelrn  6910  orderseqlem  8149  dfac4  10102  dfac2b  10110  dfacacn  10121  axdc2lem  10427  axcclem  10436  seqexw  14049  plusffval  18699  grpsubfval  19045  mulgfval  19130  staffval  20944  scaffval  21001  lpival  21492  ipffval  21798  nmfval  24745  tcphex  25376  tchnmfval  25387  rrnval  38478  lsatset  39764  fvnonrel  44323