Theorem bnj1154 31067

Description: Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1154	|- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> E. x e. B A. y e. B -. y R x )

Distinct variable groups:   x, A, y   x, B, y   x, R, y

Proof of Theorem bnj1154

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj658 30821	. 2 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> ( R Fr A /\ B C_ A /\ B =/= (/) ) )
2		elisset 3215	. . . . 5 |- ( B e. _V -> E. b b = B )
3	2	bnj708 30826	. . . 4 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> E. b b = B )
4		df-fr 5073	. . . . . . . 8 |- ( R Fr A <-> A. b ( ( b C_ A /\ b =/= (/) ) -> E. x e. b A. y e. b -. y R x ) )
5	4	biimpi 206	. . . . . . 7 |- ( R Fr A -> A. b ( ( b C_ A /\ b =/= (/) ) -> E. x e. b A. y e. b -. y R x ) )
6	5	19.21bi 2059	. . . . . 6 |- ( R Fr A -> ( ( b C_ A /\ b =/= (/) ) -> E. x e. b A. y e. b -. y R x ) )
7	6	3impib 1262	. . . . 5 |- ( ( R Fr A /\ b C_ A /\ b =/= (/) ) -> E. x e. b A. y e. b -. y R x )
8		sseq1 3626	. . . . . . 7 |- ( b = B -> ( b C_ A <-> B C_ A ) )
9		neeq1 2856	. . . . . . 7 |- ( b = B -> ( b =/= (/) <-> B =/= (/) ) )
10	8, 9	3anbi23d 1402	. . . . . 6 |- ( b = B -> ( ( R Fr A /\ b C_ A /\ b =/= (/) ) <-> ( R Fr A /\ B C_ A /\ B =/= (/) ) ) )
11		raleq 3138	. . . . . . 7 |- ( b = B -> ( A. y e. b -. y R x <-> A. y e. B -. y R x ) )
12	11	rexeqbi1dv 3147	. . . . . 6 |- ( b = B -> ( E. x e. b A. y e. b -. y R x <-> E. x e. B A. y e. B -. y R x ) )
13	10, 12	imbi12d 334	. . . . 5 |- ( b = B -> ( ( ( R Fr A /\ b C_ A /\ b =/= (/) ) -> E. x e. b A. y e. b -. y R x ) <-> ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B A. y e. B -. y R x ) ) )
14	7, 13	mpbii 223	. . . 4 |- ( b = B -> ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B A. y e. B -. y R x ) )
15	3, 14	bnj593 30815	. . 3 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> E. b ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B A. y e. B -. y R x ) )
16	15	bnj937 30842	. 2 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B A. y e. B -. y R x ) )
17	1, 16	mpd 15	1 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) /\ B e. _V ) -> E. x e. B A. y e. B -. y R x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070   /\ w-bnj17 30752

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

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-v 3202  df-in 3581  df-ss 3588  df-fr 5073  df-bnj17 30753

This theorem is referenced by:  bnj1190  31076