Theorem fvrn0 6216

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 ) e. ( ran F u. { (/) } )

Proof of Theorem fvrn0

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( ( F ` X ) = (/) -> ( F ` X ) = (/) )
2		ssun2 3777	. . . 4 |- { (/) } C_ ( ran F u. { (/) } )
3		0ex 4790	. . . . 5 |- (/) e. _V
4	3	snid 4208	. . . 4 |- (/) e. { (/) }
5	2, 4	sselii 3600	. . 3 |- (/) e. ( ran F u. { (/) } )
6	1, 5	syl6eqel 2709	. 2 |- ( ( F ` X ) = (/) -> ( F ` X ) e. ( ran F u. { (/) } ) )
7		ssun1 3776	. . 3 |- ran F C_ ( ran F u. { (/) } )
8		fvprc 6185	. . . . 5 |- ( -. X e. _V -> ( F ` X ) = (/) )
9	8	con1i 144	. . . 4 |- ( -. ( F ` X ) = (/) -> X e. _V )
10		fvexd 6203	. . . 4 |- ( -. ( F ` X ) = (/) -> ( F ` X ) e. _V )
11		fvbr0 6215	. . . . . 6 |- ( X F ( F ` X ) \/ ( F ` X ) = (/) )
12	11	ori 390	. . . . 5 |- ( -. X F ( F ` X ) -> ( F ` X ) = (/) )
13	12	con1i 144	. . . 4 |- ( -. ( F ` X ) = (/) -> X F ( F ` X ) )
14		brelrng 5355	. . . 4 |- ( ( X e. _V /\ ( F ` X ) e. _V /\ X F ( F ` X ) ) -> ( F ` X ) e. ran F )
15	9, 10, 13, 14	syl3anc 1326	. . 3 |- ( -. ( F ` X ) = (/) -> ( F ` X ) e. ran F )
16	7, 15	sseldi 3601	. 2 |- ( -. ( F ` X ) = (/) -> ( F ` X ) e. ( ran F u. { (/) } ) )
17	6, 16	pm2.61i 176	1 |- ( F ` X ) e. ( ran F u. { (/) } )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   class class class wbr 4653   ran crn 5115   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvssunirn  6217  dfac4  8945  dfac2  8953  dfacacn  8963  axdc2lem  9270  axcclem  9279  plusffval  17247  staffval  18847  scaffval  18881  lpival  19245  ipffval  19993  nmfval  22393  tchex  23016  tchnmfval  23027  orderseqlem  31749  rrnval  33626  lsatset  34277  fvnonrel  37903