Theorem bnj1491 31125

Description: Technical lemma for bnj60 31130. 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
bnj1491	|- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )

Distinct variable groups:   A, f   f, G   R, f   x, f

Allowed substitution hints:   ps( x, y, f, d)   ch( x, y, f, d)   ta( x, y, f, d)   A( x, y, d)   B( x, y, f, d)   C( x, y, f, d)   D( x, y, f, d)   P( x, y, f, d)   Q( x, y, f, d)   R( x, y, d)   G( x, y, d)   H( x, y, f, d)   Y( x, y, f, d)   Z( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1491

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1491.13	. 2 |- ( ch -> ( Q e. C /\ dom Q = ( { x } u. trCl ( x , A , R ) ) ) )
2		bnj1491.1	. . . . 5 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
3		bnj1491.2	. . . . 5 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
4		bnj1491.3	. . . . 5 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
5		bnj1491.4	. . . . 5 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
6		bnj1491.5	. . . . 5 |- D = { x e. A | -. E. f ta }
7		bnj1491.6	. . . . 5 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
8		bnj1491.7	. . . . 5 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
9		bnj1491.8	. . . . 5 |- ( ta' <-> [. y / x ]. ta )
10		bnj1491.9	. . . . 5 |- H = { f | E. y e. pred ( x , A , R ) ta' }
11		bnj1491.10	. . . . 5 |- P = U. H
12		bnj1491.11	. . . . 5 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
13		bnj1491.12	. . . . 5 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
14	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	bnj1466 31121	. . . 4 |- ( w e. Q -> A. f w e. Q )
15	14	nfcii 2755	. . 3 |- F/_ f Q
16	4	bnj1317 30892	. . . . . 6 |- ( w e. C -> A. f w e. C )
17	16	nfcii 2755	. . . . 5 |- F/_ f C
18	15, 17	nfel 2777	. . . 4 |- F/ f Q e. C
19	15	nfdm 5367	. . . . 5 |- F/_ f dom Q
20	19	nfeq1 2778	. . . 4 |- F/ f dom Q = ( { x } u. trCl ( x , A , R ) )
21	18, 20	nfan 1828	. . 3 |- F/ f ( Q e. C /\ dom Q = ( { x } u. trCl ( x , A , R ) ) )
22		eleq1 2689	. . . 4 |- ( f = Q -> ( f e. C <-> Q e. C ) )
23		dmeq 5324	. . . . 5 |- ( f = Q -> dom f = dom Q )
24	23	eqeq1d 2624	. . . 4 |- ( f = Q -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom Q = ( { x } u. trCl ( x , A , R ) ) ) )
25	22, 24	anbi12d 747	. . 3 |- ( 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 3289	. 2 |- ( 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	1, 26	mpan9 486	1 |- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-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-dm 5124  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1312  31126