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Theorem bnj1124 31056
Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1124.4 |- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1124.5 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
Assertion
Ref Expression
bnj1124 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )
Proof of Theorem bnj1124
Dummy variables f i j n y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 251 . 2 |- ( ( f ` (/) ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) )
2 biid 251 . 2 |- ( A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3 biid 251 . 2 |- ( ( n e. ( om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> ( n e. ( om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
4 bnj1124.4 . 2 |- ( th <-> ( R FrSe A /\ X e. A ) )
5 bnj1124.5 . 2 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
6 biid 251 . 2 |- ( ( i e. n /\ z e. ( f ` i ) ) <-> ( i e. n /\ z e. ( f ` i ) ) )
7 eqid 2622 . 2 |- ( om \ { (/) } ) = ( om \ { (/) } )
8 eqid 2622 . 2 |- { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } = { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) }
9 biid 251 . 2 |- ( ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) <-> ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) )
10 biid 251 . 2 |- ( A. j e. n ( j _E i -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) <-> A. j e. n ( j _E i -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) )
11 biid 251 . 2 |- ( [. j / i ]. ( f ` (/) ) = pred ( X , A , R ) <-> [. j / i ]. ( f ` (/) ) = pred ( X , A , R ) )
12 biid 251 . 2 |- ( [. j / i ]. A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> [. j / i ]. A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
13 biid 251 . 2 |- ( [. j / i ]. ( n e. ( om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> [. j / i ]. ( n e. ( om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14 biid 251 . 2 |- ( [. j / i ]. th <-> [. j / i ]. th )
15 biid 251 . 2 |- ( [. j / i ]. ta <-> [. j / i ]. ta )
16 biid 251 . 2 |- ( [. j / i ]. ( i e. n /\ z e. ( f ` i ) ) <-> [. j / i ]. ( i e. n /\ z e. ( f ` i ) ) )
17 biid 251 . 2 |- ( [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) <-> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) )
18 biid 251 . 2 |- ( ( ( j e. n /\ j _E i ) -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) <-> ( ( j e. n /\ j _E i ) -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) )
19 biid 251 . 2 |- ( ( i e. n /\ ( ( j e. n /\ j _E i ) -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) /\ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) <-> ( i e. n /\ ( ( j e. n /\ j _E i ) -> [. j / i ]. ( ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) -> ( f ` i ) C_ B ) ) /\ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ i e. dom f ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19bnj1030 31055 1 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760   TrFow-bnj19 30762
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-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949
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-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-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj18 30761  df-bnj19 30763
This theorem is referenced by:  bnj1125  31060  bnj1136  31065  bnj1408  31104
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