Theorem brwitnlem 7587

Description: Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)

Hypotheses

Ref	Expression
brwitnlem.r	|- R = ( `' O " ( _V \ 1o ) )
brwitnlem.o	|- O Fn X

Assertion

Ref	Expression
brwitnlem	|- ( A R B <-> ( A O B ) =/= (/) )

Proof of Theorem brwitnlem

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- ( O ` <. A , B >. ) e. _V
2		dif1o 7580	. . . . 5 |- ( ( O ` <. A , B >. ) e. ( _V \ 1o ) <-> ( ( O ` <. A , B >. ) e. _V /\ ( O ` <. A , B >. ) =/= (/) ) )
3	1, 2	mpbiran 953	. . . 4 |- ( ( O ` <. A , B >. ) e. ( _V \ 1o ) <-> ( O ` <. A , B >. ) =/= (/) )
4	3	anbi2i 730	. . 3 |- ( ( <. A , B >. e. X /\ ( O ` <. A , B >. ) e. ( _V \ 1o ) ) <-> ( <. A , B >. e. X /\ ( O ` <. A , B >. ) =/= (/) ) )
5		brwitnlem.o	. . . 4 |- O Fn X
6		elpreima 6337	. . . 4 |- ( O Fn X -> ( <. A , B >. e. ( `' O " ( _V \ 1o ) ) <-> ( <. A , B >. e. X /\ ( O ` <. A , B >. ) e. ( _V \ 1o ) ) ) )
7	5, 6	ax-mp 5	. . 3 |- ( <. A , B >. e. ( `' O " ( _V \ 1o ) ) <-> ( <. A , B >. e. X /\ ( O ` <. A , B >. ) e. ( _V \ 1o ) ) )
8		ndmfv 6218	. . . . . 6 |- ( -. <. A , B >. e. dom O -> ( O ` <. A , B >. ) = (/) )
9	8	necon1ai 2821	. . . . 5 |- ( ( O ` <. A , B >. ) =/= (/) -> <. A , B >. e. dom O )
10		fndm 5990	. . . . . 6 |- ( O Fn X -> dom O = X )
11	5, 10	ax-mp 5	. . . . 5 |- dom O = X
12	9, 11	syl6eleq 2711	. . . 4 |- ( ( O ` <. A , B >. ) =/= (/) -> <. A , B >. e. X )
13	12	pm4.71ri 665	. . 3 |- ( ( O ` <. A , B >. ) =/= (/) <-> ( <. A , B >. e. X /\ ( O ` <. A , B >. ) =/= (/) ) )
14	4, 7, 13	3bitr4i 292	. 2 |- ( <. A , B >. e. ( `' O " ( _V \ 1o ) ) <-> ( O ` <. A , B >. ) =/= (/) )
15		brwitnlem.r	. . . 4 |- R = ( `' O " ( _V \ 1o ) )
16	15	breqi 4659	. . 3 |- ( A R B <-> A ( `' O " ( _V \ 1o ) ) B )
17		df-br 4654	. . 3 |- ( A ( `' O " ( _V \ 1o ) ) B <-> <. A , B >. e. ( `' O " ( _V \ 1o ) ) )
18	16, 17	bitri 264	. 2 |- ( A R B <-> <. A , B >. e. ( `' O " ( _V \ 1o ) ) )
19		df-ov 6653	. . 3 |- ( A O B ) = ( O ` <. A , B >. )
20	19	neeq1i 2858	. 2 |- ( ( A O B ) =/= (/) <-> ( O ` <. A , B >. ) =/= (/) )
21	14, 18, 20	3bitr4i 292	1 |- ( A R B <-> ( A O B ) =/= (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   (/)c0 3915   <.cop 4183   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1oc1o 7553

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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-1o 7560

This theorem is referenced by:  brgic  17711  brric  18744  brlmic  19068  hmph  21579