Theorem brwdom 8472

Description: Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
brwdom	|- ( Y e. V -> ( X ~<_* Y <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) )

Distinct variable groups:   z, X   z, Y

Allowed substitution hint:   V( z)

Proof of Theorem brwdom

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

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( Y e. V -> Y e. _V )
2		relwdom 8471	. . . . 5 |- Rel ~<_*
3	2	brrelexi 5158	. . . 4 |- ( X ~<_* Y -> X e. _V )
4	3	a1i 11	. . 3 |- ( Y e. _V -> ( X ~<_* Y -> X e. _V ) )
5		0ex 4790	. . . . . 6 |- (/) e. _V
6		eleq1a 2696	. . . . . 6 |- ( (/) e. _V -> ( X = (/) -> X e. _V ) )
7	5, 6	ax-mp 5	. . . . 5 |- ( X = (/) -> X e. _V )
8		forn 6118	. . . . . . 7 |- ( z : Y -onto-> X -> ran z = X )
9		vex 3203	. . . . . . . 8 |- z e. _V
10	9	rnex 7100	. . . . . . 7 |- ran z e. _V
11	8, 10	syl6eqelr 2710	. . . . . 6 |- ( z : Y -onto-> X -> X e. _V )
12	11	exlimiv 1858	. . . . 5 |- ( E. z z : Y -onto-> X -> X e. _V )
13	7, 12	jaoi 394	. . . 4 |- ( ( X = (/) \/ E. z z : Y -onto-> X ) -> X e. _V )
14	13	a1i 11	. . 3 |- ( Y e. _V -> ( ( X = (/) \/ E. z z : Y -onto-> X ) -> X e. _V ) )
15		eqeq1 2626	. . . . . 6 |- ( x = X -> ( x = (/) <-> X = (/) ) )
16		foeq3 6113	. . . . . . 7 |- ( x = X -> ( z : y -onto-> x <-> z : y -onto-> X ) )
17	16	exbidv 1850	. . . . . 6 |- ( x = X -> ( E. z z : y -onto-> x <-> E. z z : y -onto-> X ) )
18	15, 17	orbi12d 746	. . . . 5 |- ( x = X -> ( ( x = (/) \/ E. z z : y -onto-> x ) <-> ( X = (/) \/ E. z z : y -onto-> X ) ) )
19		foeq2 6112	. . . . . . 7 |- ( y = Y -> ( z : y -onto-> X <-> z : Y -onto-> X ) )
20	19	exbidv 1850	. . . . . 6 |- ( y = Y -> ( E. z z : y -onto-> X <-> E. z z : Y -onto-> X ) )
21	20	orbi2d 738	. . . . 5 |- ( y = Y -> ( ( X = (/) \/ E. z z : y -onto-> X ) <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) )
22		df-wdom 8464	. . . . 5 |- ~<_* = { <. x , y >. | ( x = (/) \/ E. z z : y -onto-> x ) }
23	18, 21, 22	brabg 4994	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X ~<_* Y <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) )
24	23	expcom 451	. . 3 |- ( Y e. _V -> ( X e. _V -> ( X ~<_* Y <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) ) )
25	4, 14, 24	pm5.21ndd 369	. 2 |- ( Y e. _V -> ( X ~<_* Y <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) )
26	1, 25	syl 17	1 |- ( Y e. V -> ( X ~<_* Y <-> ( X = (/) \/ E. z z : Y -onto-> X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ran crn 5115   -onto->wfo 5886   ~<_^* cwdom 8462

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-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fn 5891  df-fo 5894  df-wdom 8464

This theorem is referenced by:  brwdomi  8473  brwdomn0  8474  0wdom  8475  fowdom  8476  domwdom  8479  wdomnumr  8887