Theorem bnj1421 31110

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
bnj1421.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1421.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1421.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1421.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1421.5	|- D = { x e. A | -. E. f ta }
bnj1421.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1421.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1421.8	|- ( ta' <-> [. y / x ]. ta )
bnj1421.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1421.10	|- P = U. H
bnj1421.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1421.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1421.13	|- ( ch -> Fun P )
bnj1421.14	|- ( ch -> dom Q = ( { x } u. trCl ( x , A , R ) ) )
bnj1421.15	|- ( ch -> dom P = trCl ( x , A , R ) )

Assertion

Ref	Expression
bnj1421	|- ( ch -> Fun Q )

Distinct variable groups:   x, A   x, R

Allowed substitution hints:   ps( x, y, f, d)   ch( x, y, f, d)   ta( x, y, f, d)   A( y, f, 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( y, f, d)   G( x, y, f, d)   H( x, y, f, d)   Y( x, y, f, d)   Z( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1421

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1421.13	. . . 4 |- ( ch -> Fun P )
2		vex 3203	. . . . 5 |- x e. _V
3		fvex 6201	. . . . 5 |- ( G ` Z ) e. _V
4	2, 3	funsn 5939	. . . 4 |- Fun { <. x , ( G ` Z ) >. }
5	1, 4	jctir 561	. . 3 |- ( ch -> ( Fun P /\ Fun { <. x , ( G ` Z ) >. } ) )
6		bnj1421.15	. . . . 5 |- ( ch -> dom P = trCl ( x , A , R ) )
7	3	dmsnop 5609	. . . . . 6 |- dom { <. x , ( G ` Z ) >. } = { x }
8	7	a1i 11	. . . . 5 |- ( ch -> dom { <. x , ( G ` Z ) >. } = { x } )
9	6, 8	ineq12d 3815	. . . 4 |- ( ch -> ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = ( trCl ( x , A , R ) i^i { x } ) )
10		bnj1421.7	. . . . . . 7 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
11		bnj1421.6	. . . . . . . 8 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
12	11	simplbi 476	. . . . . . 7 |- ( ps -> R FrSe A )
13	10, 12	bnj835 30829	. . . . . 6 |- ( ch -> R FrSe A )
14		biid 251	. . . . . . . 8 |- ( R FrSe A <-> R FrSe A )
15		biid 251	. . . . . . . 8 |- ( -. x e. trCl ( x , A , R ) <-> -. x e. trCl ( x , A , R ) )
16		biid 251	. . . . . . . 8 |- ( A. z e. A ( z R x -> [. z / x ]. -. x e. trCl ( x , A , R ) ) <-> A. z e. A ( z R x -> [. z / x ]. -. x e. trCl ( x , A , R ) ) )
17		biid 251	. . . . . . . 8 |- ( ( R FrSe A /\ x e. A /\ A. z e. A ( z R x -> [. z / x ]. -. x e. trCl ( x , A , R ) ) ) <-> ( R FrSe A /\ x e. A /\ A. z e. A ( z R x -> [. z / x ]. -. x e. trCl ( x , A , R ) ) ) )
18		eqid 2622	. . . . . . . 8 |- ( pred ( x , A , R ) u. U_ z e. pred ( x , A , R ) trCl ( z , A , R ) ) = ( pred ( x , A , R ) u. U_ z e. pred ( x , A , R ) trCl ( z , A , R ) )
19	14, 15, 16, 17, 18	bnj1417 31109	. . . . . . 7 |- ( R FrSe A -> A. x e. A -. x e. trCl ( x , A , R ) )
20		disjsn 4246	. . . . . . . 8 |- ( ( trCl ( x , A , R ) i^i { x } ) = (/) <-> -. x e. trCl ( x , A , R ) )
21	20	ralbii 2980	. . . . . . 7 |- ( A. x e. A ( trCl ( x , A , R ) i^i { x } ) = (/) <-> A. x e. A -. x e. trCl ( x , A , R ) )
22	19, 21	sylibr 224	. . . . . 6 |- ( R FrSe A -> A. x e. A ( trCl ( x , A , R ) i^i { x } ) = (/) )
23	13, 22	syl 17	. . . . 5 |- ( ch -> A. x e. A ( trCl ( x , A , R ) i^i { x } ) = (/) )
24		bnj1421.5	. . . . . 6 |- D = { x e. A | -. E. f ta }
25	24, 10	bnj1212 30870	. . . . 5 |- ( ch -> x e. A )
26	23, 25	bnj1294 30888	. . . 4 |- ( ch -> ( trCl ( x , A , R ) i^i { x } ) = (/) )
27	9, 26	eqtrd 2656	. . 3 |- ( ch -> ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = (/) )
28		funun 5932	. . 3 |- ( ( ( Fun P /\ Fun { <. x , ( G ` Z ) >. } ) /\ ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = (/) ) -> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
29	5, 27, 28	syl2anc 693	. 2 |- ( ch -> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
30		bnj1421.12	. . 3 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
31	30	funeqi 5909	. 2 |- ( Fun Q <-> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
32	29, 31	sylibr 224	1 |- ( ch -> Fun Q )

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   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1312  31126