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Theorem relexpss1d 36343
Description: The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
Hypotheses
Ref Expression
relexpss1d.a  |-  ( ph  ->  A  C_  B )
relexpss1d.b  |-  ( ph  ->  B  e.  _V )
relexpss1d.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
relexpss1d  |-  ( ph  ->  ( A ^r  N )  C_  ( B ^r  N ) )

Proof of Theorem relexpss1d
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relexpss1d.n . . 3  |-  ( ph  ->  N  e.  NN0 )
2 elnn0 10905 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
31, 2sylib 201 . 2  |-  ( ph  ->  ( N  e.  NN  \/  N  =  0
) )
4 oveq2 6328 . . . . . 6  |-  ( x  =  1  ->  ( A ^r  x )  =  ( A ^r  1 ) )
5 oveq2 6328 . . . . . 6  |-  ( x  =  1  ->  ( B ^r  x )  =  ( B ^r  1 ) )
64, 5sseq12d 3473 . . . . 5  |-  ( x  =  1  ->  (
( A ^r 
x )  C_  ( B ^r  x )  <-> 
( A ^r 
1 )  C_  ( B ^r  1 ) ) )
76imbi2d 322 . . . 4  |-  ( x  =  1  ->  (
( ph  ->  ( A ^r  x ) 
C_  ( B ^r  x ) )  <-> 
( ph  ->  ( A ^r  1 ) 
C_  ( B ^r  1 ) ) ) )
8 oveq2 6328 . . . . . 6  |-  ( x  =  y  ->  ( A ^r  x )  =  ( A ^r  y ) )
9 oveq2 6328 . . . . . 6  |-  ( x  =  y  ->  ( B ^r  x )  =  ( B ^r  y ) )
108, 9sseq12d 3473 . . . . 5  |-  ( x  =  y  ->  (
( A ^r 
x )  C_  ( B ^r  x )  <-> 
( A ^r 
y )  C_  ( B ^r  y ) ) )
1110imbi2d 322 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ( A ^r  x ) 
C_  ( B ^r  x ) )  <-> 
( ph  ->  ( A ^r  y ) 
C_  ( B ^r  y ) ) ) )
12 oveq2 6328 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  ( A ^r  x )  =  ( A ^r  ( y  +  1 ) ) )
13 oveq2 6328 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  ( B ^r  x )  =  ( B ^r  ( y  +  1 ) ) )
1412, 13sseq12d 3473 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
( A ^r 
x )  C_  ( B ^r  x )  <-> 
( A ^r 
( y  +  1 ) )  C_  ( B ^r  ( y  +  1 ) ) ) )
1514imbi2d 322 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( ph  ->  ( A ^r  x ) 
C_  ( B ^r  x ) )  <-> 
( ph  ->  ( A ^r  ( y  +  1 ) ) 
C_  ( B ^r  ( y  +  1 ) ) ) ) )
16 oveq2 6328 . . . . . 6  |-  ( x  =  N  ->  ( A ^r  x )  =  ( A ^r  N ) )
17 oveq2 6328 . . . . . 6  |-  ( x  =  N  ->  ( B ^r  x )  =  ( B ^r  N ) )
1816, 17sseq12d 3473 . . . . 5  |-  ( x  =  N  ->  (
( A ^r 
x )  C_  ( B ^r  x )  <-> 
( A ^r  N )  C_  ( B ^r  N ) ) )
1918imbi2d 322 . . . 4  |-  ( x  =  N  ->  (
( ph  ->  ( A ^r  x ) 
C_  ( B ^r  x ) )  <-> 
( ph  ->  ( A ^r  N ) 
C_  ( B ^r  N ) ) ) )
20 relexpss1d.a . . . . 5  |-  ( ph  ->  A  C_  B )
21 relexpss1d.b . . . . . . 7  |-  ( ph  ->  B  e.  _V )
2221, 20ssexd 4566 . . . . . 6  |-  ( ph  ->  A  e.  _V )
2322relexp1d 13149 . . . . 5  |-  ( ph  ->  ( A ^r 
1 )  =  A )
2421relexp1d 13149 . . . . 5  |-  ( ph  ->  ( B ^r 
1 )  =  B )
2520, 23, 243sstr4d 3487 . . . 4  |-  ( ph  ->  ( A ^r 
1 )  C_  ( B ^r  1 ) )
26 simp3 1016 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
( A ^r 
y )  C_  ( B ^r  y ) )
27203ad2ant2 1036 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  ->  A  C_  B )
2826, 27coss12d 13091 . . . . . . 7  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
( ( A ^r  y )  o.  A )  C_  (
( B ^r 
y )  o.  B
) )
29223ad2ant2 1036 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  ->  A  e.  _V )
30 simp1 1014 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
y  e.  NN )
31 relexpsucnnr 13143 . . . . . . . 8  |-  ( ( A  e.  _V  /\  y  e.  NN )  ->  ( A ^r 
( y  +  1 ) )  =  ( ( A ^r 
y )  o.  A
) )
3229, 30, 31syl2anc 671 . . . . . . 7  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
( A ^r 
( y  +  1 ) )  =  ( ( A ^r 
y )  o.  A
) )
33213ad2ant2 1036 . . . . . . . 8  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  ->  B  e.  _V )
34 relexpsucnnr 13143 . . . . . . . 8  |-  ( ( B  e.  _V  /\  y  e.  NN )  ->  ( B ^r 
( y  +  1 ) )  =  ( ( B ^r 
y )  o.  B
) )
3533, 30, 34syl2anc 671 . . . . . . 7  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
( B ^r 
( y  +  1 ) )  =  ( ( B ^r 
y )  o.  B
) )
3628, 32, 353sstr4d 3487 . . . . . 6  |-  ( ( y  e.  NN  /\  ph 
/\  ( A ^r  y )  C_  ( B ^r 
y ) )  -> 
( A ^r 
( y  +  1 ) )  C_  ( B ^r  ( y  +  1 ) ) )
37363exp 1214 . . . . 5  |-  ( y  e.  NN  ->  ( ph  ->  ( ( A ^r  y ) 
C_  ( B ^r  y )  -> 
( A ^r 
( y  +  1 ) )  C_  ( B ^r  ( y  +  1 ) ) ) ) )
3837a2d 29 . . . 4  |-  ( y  e.  NN  ->  (
( ph  ->  ( A ^r  y ) 
C_  ( B ^r  y ) )  ->  ( ph  ->  ( A ^r  ( y  +  1 ) )  C_  ( B ^r  ( y  +  1 ) ) ) ) )
397, 11, 15, 19, 25, 38nnind 10660 . . 3  |-  ( N  e.  NN  ->  ( ph  ->  ( A ^r  N )  C_  ( B ^r  N ) ) )
40 simpr 467 . . . . . 6  |-  ( ( N  =  0  /\ 
ph )  ->  ph )
41 dmss 5056 . . . . . . . 8  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
42 rnss 5085 . . . . . . . 8  |-  ( A 
C_  B  ->  ran  A 
C_  ran  B )
4341, 42jca 539 . . . . . . 7  |-  ( A 
C_  B  ->  ( dom  A  C_  dom  B  /\  ran  A  C_  ran  B ) )
44 unss12 3618 . . . . . . 7  |-  ( ( dom  A  C_  dom  B  /\  ran  A  C_  ran  B )  ->  ( dom  A  u.  ran  A
)  C_  ( dom  B  u.  ran  B ) )
4520, 43, 443syl 18 . . . . . 6  |-  ( ph  ->  ( dom  A  u.  ran  A )  C_  ( dom  B  u.  ran  B
) )
46 ssres2 5153 . . . . . 6  |-  ( ( dom  A  u.  ran  A )  C_  ( dom  B  u.  ran  B )  ->  (  _I  |`  ( dom  A  u.  ran  A
) )  C_  (  _I  |`  ( dom  B  u.  ran  B ) ) )
4740, 45, 463syl 18 . . . . 5  |-  ( ( N  =  0  /\ 
ph )  ->  (  _I  |`  ( dom  A  u.  ran  A ) ) 
C_  (  _I  |`  ( dom  B  u.  ran  B
) ) )
48 simpl 463 . . . . . . 7  |-  ( ( N  =  0  /\ 
ph )  ->  N  =  0 )
4948oveq2d 6336 . . . . . 6  |-  ( ( N  =  0  /\ 
ph )  ->  ( A ^r  N )  =  ( A ^r  0 ) )
50 relexp0g 13140 . . . . . . 7  |-  ( A  e.  _V  ->  ( A ^r  0 )  =  (  _I  |`  ( dom  A  u.  ran  A
) ) )
5140, 22, 503syl 18 . . . . . 6  |-  ( ( N  =  0  /\ 
ph )  ->  ( A ^r  0 )  =  (  _I  |`  ( dom  A  u.  ran  A
) ) )
5249, 51eqtrd 2496 . . . . 5  |-  ( ( N  =  0  /\ 
ph )  ->  ( A ^r  N )  =  (  _I  |`  ( dom  A  u.  ran  A
) ) )
5348oveq2d 6336 . . . . . 6  |-  ( ( N  =  0  /\ 
ph )  ->  ( B ^r  N )  =  ( B ^r  0 ) )
54 relexp0g 13140 . . . . . . 7  |-  ( B  e.  _V  ->  ( B ^r  0 )  =  (  _I  |`  ( dom  B  u.  ran  B
) ) )
5540, 21, 543syl 18 . . . . . 6  |-  ( ( N  =  0  /\ 
ph )  ->  ( B ^r  0 )  =  (  _I  |`  ( dom  B  u.  ran  B
) ) )
5653, 55eqtrd 2496 . . . . 5  |-  ( ( N  =  0  /\ 
ph )  ->  ( B ^r  N )  =  (  _I  |`  ( dom  B  u.  ran  B
) ) )
5747, 52, 563sstr4d 3487 . . . 4  |-  ( ( N  =  0  /\ 
ph )  ->  ( A ^r  N ) 
C_  ( B ^r  N ) )
5857ex 440 . . 3  |-  ( N  =  0  ->  ( ph  ->  ( A ^r  N )  C_  ( B ^r  N ) ) )
5939, 58jaoi 385 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ph  ->  ( A ^r  N )  C_  ( B ^r  N ) ) )
603, 59mpcom 37 1  |-  ( ph  ->  ( A ^r  N )  C_  ( B ^r  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   _Vcvv 3057    u. cun 3414    C_ wss 3416    _I cid 4766   dom cdm 4856   ran crn 4857    |` cres 4858    o. ccom 4860  (class class class)co 6320   0cc0 9570   1c1 9571    + caddc 9573   NNcn 10642   NN0cn0 10903   ^r crelexp 13138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-n0 10904  df-z 10972  df-uz 11194  df-seq 12252  df-relexp 13139
This theorem is referenced by:  corcltrcl  36377  cotrclrcl  36380
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