Theorem inisegn0 5497

Description: Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Assertion

Ref	Expression
inisegn0	|- ( A e. ran F <-> ( `' F " { A } ) =/= (/) )

Proof of Theorem inisegn0

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. ran F -> A e. _V )
2		snprc 4253	. . . . . 6 |- ( -. A e. _V <-> { A } = (/) )
3	2	biimpi 206	. . . . 5 |- ( -. A e. _V -> { A } = (/) )
4	3	imaeq2d 5466	. . . 4 |- ( -. A e. _V -> ( `' F " { A } ) = ( `' F " (/) ) )
5		ima0 5481	. . . 4 |- ( `' F " (/) ) = (/)
6	4, 5	syl6eq 2672	. . 3 |- ( -. A e. _V -> ( `' F " { A } ) = (/) )
7	6	necon1ai 2821	. 2 |- ( ( `' F " { A } ) =/= (/) -> A e. _V )
8		eleq1 2689	. . 3 |- ( a = A -> ( a e. ran F <-> A e. ran F ) )
9		sneq 4187	. . . . 5 |- ( a = A -> { a } = { A } )
10	9	imaeq2d 5466	. . . 4 |- ( a = A -> ( `' F " { a } ) = ( `' F " { A } ) )
11	10	neeq1d 2853	. . 3 |- ( a = A -> ( ( `' F " { a } ) =/= (/) <-> ( `' F " { A } ) =/= (/) ) )
12		abn0 3954	. . . 4 |- ( { b | b F a } =/= (/) <-> E. b b F a )
13		vex 3203	. . . . . 6 |- a e. _V
14		iniseg 5496	. . . . . 6 |- ( a e. _V -> ( `' F " { a } ) = { b | b F a } )
15	13, 14	ax-mp 5	. . . . 5 |- ( `' F " { a } ) = { b | b F a }
16	15	neeq1i 2858	. . . 4 |- ( ( `' F " { a } ) =/= (/) <-> { b | b F a } =/= (/) )
17	13	elrn 5366	. . . 4 |- ( a e. ran F <-> E. b b F a )
18	12, 16, 17	3bitr4ri 293	. . 3 |- ( a e. ran F <-> ( `' F " { a } ) =/= (/) )
19	8, 11, 18	vtoclbg 3267	. 2 |- ( A e. _V -> ( A e. ran F <-> ( `' F " { A } ) =/= (/) ) )
20	1, 7, 19	pm5.21nii 368	1 |- ( A e. ran F <-> ( `' F " { A } ) =/= (/) )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   class class class wbr 4653   `'ccnv 5113   ran crn 5115   "cima 5117

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  dnnumch3lem  37616  dnnumch3  37617  wessf1ornlem  39371