bnj1491

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1491.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1491.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1491.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1491.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
	bnj1491.5	\|- D = { x e. A \| -. E. f ta }
	bnj1491.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
	bnj1491.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
	bnj1491.8	\|- ( ta' <-> [. y / x ]. ta )
	bnj1491.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
	bnj1491.10	\|- P = U. H
	bnj1491.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
	bnj1491.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
	bnj1491.13	\|- ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
Assertion	bnj1491	\|- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1491.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1491.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1491.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1491.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5		bnj1491.5	\|- D = { x e. A \| -. E. f ta }
6		bnj1491.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7		bnj1491.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8		bnj1491.8	\|- ( ta' <-> [. y / x ]. ta )
9		bnj1491.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
10		bnj1491.10	\|- P = U. H
11		bnj1491.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
12		bnj1491.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
13		bnj1491.13	\|- ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
14	1 2 3 4 5 6 7 8 9 10 11 12	bnj1466	\|- ( w e. Q -> A. f w e. Q )
15	14	nfcii	\|- F/_ f Q
16	3	bnj1317	\|- ( w e. C -> A. f w e. C )
17	16	nfcii	\|- F/_ f C
18	15 17	nfel	\|- F/ f Q e. C
19	15	nfdm	\|- F/_ f dom Q
20	19	nfeq1	\|- F/ f dom Q = ( { x } u. _trCl ( x , A , R ) )
21	18 20	nfan	\|- F/ f ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) )
22		eleq1	\|- ( f = Q -> ( f e. C <-> Q e. C ) )
23		dmeq	\|- ( f = Q -> dom f = dom Q )
24	23	eqeq1d	\|- ( f = Q -> ( dom f = ( { x } u. _trCl ( x , A , R ) ) <-> dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
25	22 24	anbi12d	\|- ( f = Q -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) ) )
26	15 21 25	spcegf	\|- ( Q e. _V -> ( ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) )
27	13 26	mpan9	\|- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )

Metamath Proof Explorer

Theorem bnj1491

Proof