Theorem fvrn0 6173

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 3755	. . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅})
3		0ex 4750	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4179	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3580	. . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅})
6	1, 5	syl6eqel 2706	. 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
7		ssun1 3754	. . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8		fvprc 6142	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9	8	con1i 144	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V)
10		fvex 6158	. . . . 5 ⊢ (𝐹‘𝑋) ∈ V
11	10	a1i 11	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V)
12		fvbr0 6172	. . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
13	12	ori 390	. . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
14	13	con1i 144	. . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋))
15		brelrng 5315	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹)
16	9, 11, 14, 15	syl3anc 1323	. . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
17	7, 16	sseldi 3581	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}))
18	6, 17	pm2.61i 176	1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})

Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ∪ cun 3553  ∅c0 3891  {csn 4148   class class class wbr 4613  ran crn 5075  ‘cfv 5847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-cnv 5082  df-dm 5084  df-rn 5085  df-iota 5810  df-fv 5855

This theorem is referenced by:  fvssunirn  6174  dfac4  8889  dfac2  8897  dfacacn  8907  axdc2lem  9214  axcclem  9223  plusffval  17168  staffval  18768  scaffval  18802  lpival  19164  ipffval  19912  nmfval  22303  tchex  22924  tchnmfval  22935  orderseqlem  31447  rrnval  33255  lsatset  33754  fvnonrel  37381