Theorem bnj1416 31107

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
bnj1416.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1416.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1416.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1416.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1416.5	|- D = { x e. A | -. E. f ta }
bnj1416.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1416.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1416.8	|- ( ta' <-> [. y / x ]. ta )
bnj1416.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1416.10	|- P = U. H
bnj1416.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1416.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1416.28	|- ( ch -> dom P = trCl ( x , A , R ) )

Assertion

Ref	Expression
bnj1416	|- ( ch -> dom Q = ( { x } u. trCl ( x , A , R ) ) )

Proof of Theorem bnj1416

Step	Hyp	Ref	Expression
1		bnj1416.12	. . . 4 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
2	1	dmeqi 5325	. . 3 |- dom Q = dom ( P u. { <. x , ( G ` Z ) >. } )
3		dmun 5331	. . 3 |- dom ( P u. { <. x , ( G ` Z ) >. } ) = ( dom P u. dom { <. x , ( G ` Z ) >. } )
4		fvex 6201	. . . . 5 |- ( G ` Z ) e. _V
5	4	dmsnop 5609	. . . 4 |- dom { <. x , ( G ` Z ) >. } = { x }
6	5	uneq2i 3764	. . 3 |- ( dom P u. dom { <. x , ( G ` Z ) >. } ) = ( dom P u. { x } )
7	2, 3, 6	3eqtri 2648	. 2 |- dom Q = ( dom P u. { x } )
8		bnj1416.28	. . . 4 |- ( ch -> dom P = trCl ( x , A , R ) )
9	8	uneq1d 3766	. . 3 |- ( ch -> ( dom P u. { x } ) = ( trCl ( x , A , R ) u. { x } ) )
10		uncom 3757	. . 3 |- ( trCl ( x , A , R ) u. { x } ) = ( { x } u. trCl ( x , A , R ) )
11	9, 10	syl6eq 2672	. 2 |- ( ch -> ( dom P u. { x } ) = ( { x } u. trCl ( x , A , R ) ) )
12	7, 11	syl5eq 2668	1 |- ( ch -> dom Q = ( { 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   [.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  ax-sep 4781  ax-nul 4789  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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1312  31126