Theorem frc 2920

Description: Property of founded relation (one direction of definition using class variables).

Hypothesis

Ref	Expression
frc.1	|- B e. V

Assertion

Ref	Expression
frc	|- ((R Fr A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yRx}) = (/))

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

Proof of Theorem frc

Step	Hyp	Ref	Expression
1		dffr2 2919	. . 3 |- (R Fr A <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z (z i^i {y | yRx}) = (/)))
2		frc.1	. . . 4 |- B e. V
3		sseq1 2082	. . . . . 6 |- (z = B -> (z (_ A <-> B (_ A))
4		neeq1 1590	. . . . . 6 |- (z = B -> (z =/= (/) <-> B =/= (/)))
5	3, 4	anbi12d 628	. . . . 5 |- (z = B -> ((z (_ A /\ z =/= (/)) <-> (B (_ A /\ B =/= (/))))
6		ineq1 2210	. . . . . . 7 |- (z = B -> (z i^i {y | yRx}) = (B i^i {y | yRx}))
7	6	eqeq1d 1483	. . . . . 6 |- (z = B -> ((z i^i {y | yRx}) = (/) <-> (B i^i {y | yRx}) = (/)))
8	7	rexeqd 1792	. . . . 5 |- (z = B -> (E.x e. z (z i^i {y | yRx}) = (/) <-> E.x e. B (B i^i {y | yRx}) = (/)))
9	5, 8	imbi12d 626	. . . 4 |- (z = B -> (((z (_ A /\ z =/= (/)) -> E.x e. z (z i^i {y | yRx}) = (/)) <-> ((B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yRx}) = (/))))
10	2, 9	cla4v 1868	. . 3 |- (A.z((z (_ A /\ z =/= (/)) -> E.x e. z (z i^i {y | yRx}) = (/)) -> ((B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yRx}) = (/)))
11	1, 10	sylbi 199	. 2 |- (R Fr A -> ((B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yRx}) = (/)))
12	11	3impib 831	1 |- ((R Fr A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yRx}) = (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  {cab 1463   =/= wne 1585  E.wrex 1646  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619   Fr wfr 2915

This theorem is referenced by:  frirr 2924  fr2nr 2925  fr3nr 2926  epfrc 2933

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-fr 2917

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