Theorem fvrn0 6918

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 4176	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 5260	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4619	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3906	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	eqeltrdi 2868	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 4175	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6882	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 148	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvexd 6905	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
11		fvbr0 6917	. . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
12	11	ori 875	. . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
13	12	con1i 148	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋))
14		brelrng 5924	. . . 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 3907	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
17	6, 16	pm2.61i 184	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∪ cun 4148  ∅c0 4277  {csn 4584   class class class wbr 5102  ran crn 5652  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6497  df-fv 6551

This theorem is used by:  fvn0fvelrn  6919  orderseqlem  8180  dfac4  10208  dfac2b  10216  dfacacn  10227  axdc2lem  10533  axcclem  10542  seqexw  14167  plusffval  18831  grpsubfval  19203  mulgfval  19288  staffval  21107  scaffval  21164  lpival  21657  ipffval  21963  nmfval  24916  tcphex  25547  tchnmfval  25558  rrnval  38870  lsatset  40156  fvnonrel  44706