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Theorem rtrclind 27487
Description: Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
rtrclind.1  |-  ( et 
->  Rel  R )
rtrclind.2  |-  ( et 
->  R  e.  _V )
rtrclind.3  |-  ( et 
->  S  e.  _V )
rtrclind.4  |-  ( et 
->  X  e.  _V )
rtrclind.5  |-  ( i  =  S  ->  ( ph 
<->  ch ) )
rtrclind.6  |-  ( i  =  x  ->  ( ph 
<->  ps ) )
rtrclind.7  |-  ( i  =  j  ->  ( ph 
<->  th ) )
rtrclind.8  |-  ( x  =  X  ->  ( ps 
<->  ta ) )
rtrclind.9  |-  ( et 
->  ch )
rtrclind.10  |-  ( et 
->  ( j R x  ->  ( th  ->  ps ) ) )
Assertion
Ref Expression
rtrclind  |-  ( et 
->  ( S ( t* `  R ) X  ->  ta )
)
Distinct variable groups:    x, R, i, j    x, S, i, j    x, X    et, x, i, j    ta, x    ps, i, j    th, i    ph, j, x    ch, i
Allowed substitution hints:    ph( i)    ps( x)    ch( x, j)    th( x, j)    ta( i, j)    X( i, j)

Proof of Theorem rtrclind
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 rtrclind.1 . . 3  |-  ( et 
->  Rel  R )
2 rtrclind.2 . . 3  |-  ( et 
->  R  e.  _V )
31, 2dfrtrcl2 27486 . 2  |-  ( et 
->  ( t* `  R )  =  ( t*rec `  R
) )
41, 2dfrtrclrec2 27481 . . . . . 6  |-  ( et 
->  ( S ( t*rec `  R ) X 
<->  E. n  e.  NN0  S ( R ^r n ) X ) )
54biimpac 486 . . . . 5  |-  ( ( S ( t*rec
`  R ) X  /\  et )  ->  E. n  e.  NN0  S ( R ^r n ) X )
6 simprl 755 . . . . . . . . . 10  |-  ( ( S ( t*rec
`  R ) X  /\  ( et  /\  ( S ( R ^r n ) X  /\  n  e.  NN0 ) ) )  ->  et )
7 simprrr 764 . . . . . . . . . 10  |-  ( ( S ( t*rec
`  R ) X  /\  ( et  /\  ( S ( R ^r n ) X  /\  n  e.  NN0 ) ) )  ->  n  e.  NN0 )
8 simprrl 763 . . . . . . . . . 10  |-  ( ( S ( t*rec
`  R ) X  /\  ( et  /\  ( S ( R ^r n ) X  /\  n  e.  NN0 ) ) )  ->  S ( R ^r n ) X )
9 rtrclind.3 . . . . . . . . . . 11  |-  ( et 
->  S  e.  _V )
10 rtrclind.4 . . . . . . . . . . 11  |-  ( et 
->  X  e.  _V )
11 rtrclind.5 . . . . . . . . . . 11  |-  ( i  =  S  ->  ( ph 
<->  ch ) )
12 rtrclind.6 . . . . . . . . . . 11  |-  ( i  =  x  ->  ( ph 
<->  ps ) )
13 rtrclind.7 . . . . . . . . . . 11  |-  ( i  =  j  ->  ( ph 
<->  th ) )
14 rtrclind.8 . . . . . . . . . . 11  |-  ( x  =  X  ->  ( ps 
<->  ta ) )
15 rtrclind.9 . . . . . . . . . . 11  |-  ( et 
->  ch )
16 rtrclind.10 . . . . . . . . . . 11  |-  ( et 
->  ( j R x  ->  ( th  ->  ps ) ) )
171, 2, 9, 10, 11, 12, 13, 14, 15, 16relexpind 27478 . . . . . . . . . 10  |-  ( et 
->  ( n  e.  NN0  ->  ( S ( R ^r n ) X  ->  ta )
) )
186, 7, 8, 17syl3c 61 . . . . . . . . 9  |-  ( ( S ( t*rec
`  R ) X  /\  ( et  /\  ( S ( R ^r n ) X  /\  n  e.  NN0 ) ) )  ->  ta )
1918anassrs 648 . . . . . . . 8  |-  ( ( ( S ( t*rec `  R ) X  /\  et )  /\  ( S ( R ^r n ) X  /\  n  e.  NN0 ) )  ->  ta )
2019expcom 435 . . . . . . 7  |-  ( ( S ( R ^r n ) X  /\  n  e.  NN0 )  ->  ( ( S ( t*rec `  R ) X  /\  et )  ->  ta )
)
2120expcom 435 . . . . . 6  |-  ( n  e.  NN0  ->  ( S ( R ^r n ) X  -> 
( ( S ( t*rec `  R
) X  /\  et )  ->  ta ) ) )
2221rexlimiv 2933 . . . . 5  |-  ( E. n  e.  NN0  S
( R ^r n ) X  -> 
( ( S ( t*rec `  R
) X  /\  et )  ->  ta ) )
235, 22mpcom 36 . . . 4  |-  ( ( S ( t*rec
`  R ) X  /\  et )  ->  ta )
2423expcom 435 . . 3  |-  ( et 
->  ( S ( t*rec `  R ) X  ->  ta ) )
25 breq 4394 . . . 4  |-  ( ( t* `  R
)  =  ( t*rec `  R )  ->  ( S ( t* `  R ) X  <->  S ( t*rec
`  R ) X ) )
2625imbi1d 317 . . 3  |-  ( ( t* `  R
)  =  ( t*rec `  R )  ->  ( ( S ( t* `  R
) X  ->  ta ) 
<->  ( S ( t*rec `  R ) X  ->  ta ) ) )
2724, 26syl5ibr 221 . 2  |-  ( ( t* `  R
)  =  ( t*rec `  R )  ->  ( et  ->  ( S ( t* `  R ) X  ->  ta ) ) )
283, 27mpcom 36 1  |-  ( et 
->  ( S ( t* `  R ) X  ->  ta )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3070   class class class wbr 4392   Rel wrel 4945   ` cfv 5518  (class class class)co 6192   NN0cn0 10682   t*crtcl 23807   ^rcrelexp 27465   t*reccrtrcl 27479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-seq 11910  df-rtrcl 23809  df-relexp 27466  df-rtrclrec 27480
This theorem is referenced by: (None)
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