Theorem eldm3 31651

Description: Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Assertion

Ref	Expression
eldm3	|- ( A e. dom B <-> ( B |` { A } ) =/= (/) )

Proof of Theorem eldm3

Dummy variables x y z p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. dom B -> A e. _V )
2		snprc 4253	. . . 4 |- ( -. A e. _V <-> { A } = (/) )
3		reseq2 5391	. . . . 5 |- ( { A } = (/) -> ( B |` { A } ) = ( B |` (/) ) )
4		res0 5400	. . . . 5 |- ( B |` (/) ) = (/)
5	3, 4	syl6eq 2672	. . . 4 |- ( { A } = (/) -> ( B |` { A } ) = (/) )
6	2, 5	sylbi 207	. . 3 |- ( -. A e. _V -> ( B |` { A } ) = (/) )
7	6	necon1ai 2821	. 2 |- ( ( B |` { A } ) =/= (/) -> A e. _V )
8		eleq1 2689	. . 3 |- ( x = A -> ( x e. dom B <-> A e. dom B ) )
9		sneq 4187	. . . . 5 |- ( x = A -> { x } = { A } )
10	9	reseq2d 5396	. . . 4 |- ( x = A -> ( B |` { x } ) = ( B |` { A } ) )
11	10	neeq1d 2853	. . 3 |- ( x = A -> ( ( B |` { x } ) =/= (/) <-> ( B |` { A } ) =/= (/) ) )
12		df-clel 2618	. . . . 5 |- ( <. x , y >. e. B <-> E. p ( p = <. x , y >. /\ p e. B ) )
13	12	exbii 1774	. . . 4 |- ( E. y <. x , y >. e. B <-> E. y E. p ( p = <. x , y >. /\ p e. B ) )
14		vex 3203	. . . . 5 |- x e. _V
15	14	eldm2 5322	. . . 4 |- ( x e. dom B <-> E. y <. x , y >. e. B )
16		n0 3931	. . . . 5 |- ( ( B |` { x } ) =/= (/) <-> E. p p e. ( B |` { x } ) )
17		elres 5435	. . . . . . 7 |- ( p e. ( B |` { x } ) <-> E. z e. { x } E. y ( p = <. z , y >. /\ <. z , y >. e. B ) )
18		eleq1 2689	. . . . . . . . . . 11 |- ( p = <. z , y >. -> ( p e. B <-> <. z , y >. e. B ) )
19	18	pm5.32i 669	. . . . . . . . . 10 |- ( ( p = <. z , y >. /\ p e. B ) <-> ( p = <. z , y >. /\ <. z , y >. e. B ) )
20		opeq1 4402	. . . . . . . . . . . 12 |- ( z = x -> <. z , y >. = <. x , y >. )
21	20	eqeq2d 2632	. . . . . . . . . . 11 |- ( z = x -> ( p = <. z , y >. <-> p = <. x , y >. ) )
22	21	anbi1d 741	. . . . . . . . . 10 |- ( z = x -> ( ( p = <. z , y >. /\ p e. B ) <-> ( p = <. x , y >. /\ p e. B ) ) )
23	19, 22	syl5bbr 274	. . . . . . . . 9 |- ( z = x -> ( ( p = <. z , y >. /\ <. z , y >. e. B ) <-> ( p = <. x , y >. /\ p e. B ) ) )
24	23	exbidv 1850	. . . . . . . 8 |- ( z = x -> ( E. y ( p = <. z , y >. /\ <. z , y >. e. B ) <-> E. y ( p = <. x , y >. /\ p e. B ) ) )
25	14, 24	rexsn 4223	. . . . . . 7 |- ( E. z e. { x } E. y ( p = <. z , y >. /\ <. z , y >. e. B ) <-> E. y ( p = <. x , y >. /\ p e. B ) )
26	17, 25	bitri 264	. . . . . 6 |- ( p e. ( B |` { x } ) <-> E. y ( p = <. x , y >. /\ p e. B ) )
27	26	exbii 1774	. . . . 5 |- ( E. p p e. ( B |` { x } ) <-> E. p E. y ( p = <. x , y >. /\ p e. B ) )
28		excom 2042	. . . . 5 |- ( E. p E. y ( p = <. x , y >. /\ p e. B ) <-> E. y E. p ( p = <. x , y >. /\ p e. B ) )
29	16, 27, 28	3bitri 286	. . . 4 |- ( ( B |` { x } ) =/= (/) <-> E. y E. p ( p = <. x , y >. /\ p e. B ) )
30	13, 15, 29	3bitr4i 292	. . 3 |- ( x e. dom B <-> ( B |` { x } ) =/= (/) )
31	8, 11, 30	vtoclbg 3267	. 2 |- ( A e. _V -> ( A e. dom B <-> ( B |` { A } ) =/= (/) ) )
32	1, 7, 31	pm5.21nii 368	1 |- ( A e. dom B <-> ( B |` { A } ) =/= (/) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116

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-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-rel 5121  df-dm 5124  df-res 5126

This theorem is referenced by:  elrn3  31652