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Theorem relexpnndm 12876
Description: The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpnndm  |-  ( ( N  e.  NN  /\  R  e.  V )  ->  dom  ( R ^r  N )  C_  dom  R )

Proof of Theorem relexpnndm
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6204 . . . . . 6  |-  ( n  =  1  ->  ( R ^r  n )  =  ( R ^r  1 ) )
21dmeqd 5118 . . . . 5  |-  ( n  =  1  ->  dom  ( R ^r 
n )  =  dom  ( R ^r 
1 ) )
32sseq1d 3444 . . . 4  |-  ( n  =  1  ->  ( dom  ( R ^r 
n )  C_  dom  R  <->  dom  ( R ^r 
1 )  C_  dom  R ) )
43imbi2d 314 . . 3  |-  ( n  =  1  ->  (
( R  e.  V  ->  dom  ( R ^r  n )  C_  dom  R )  <->  ( R  e.  V  ->  dom  ( R ^r  1 ) 
C_  dom  R )
) )
5 oveq2 6204 . . . . . 6  |-  ( n  =  m  ->  ( R ^r  n )  =  ( R ^r  m ) )
65dmeqd 5118 . . . . 5  |-  ( n  =  m  ->  dom  ( R ^r 
n )  =  dom  ( R ^r 
m ) )
76sseq1d 3444 . . . 4  |-  ( n  =  m  ->  ( dom  ( R ^r 
n )  C_  dom  R  <->  dom  ( R ^r 
m )  C_  dom  R ) )
87imbi2d 314 . . 3  |-  ( n  =  m  ->  (
( R  e.  V  ->  dom  ( R ^r  n )  C_  dom  R )  <->  ( R  e.  V  ->  dom  ( R ^r  m ) 
C_  dom  R )
) )
9 oveq2 6204 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( R ^r  n )  =  ( R ^r  ( m  + 
1 ) ) )
109dmeqd 5118 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  dom  ( R ^r 
n )  =  dom  ( R ^r 
( m  +  1 ) ) )
1110sseq1d 3444 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( dom  ( R ^r 
n )  C_  dom  R  <->  dom  ( R ^r 
( m  +  1 ) )  C_  dom  R ) )
1211imbi2d 314 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
( R  e.  V  ->  dom  ( R ^r  n )  C_  dom  R )  <->  ( R  e.  V  ->  dom  ( R ^r  ( m  +  1 ) ) 
C_  dom  R )
) )
13 oveq2 6204 . . . . . 6  |-  ( n  =  N  ->  ( R ^r  n )  =  ( R ^r  N ) )
1413dmeqd 5118 . . . . 5  |-  ( n  =  N  ->  dom  ( R ^r 
n )  =  dom  ( R ^r  N ) )
1514sseq1d 3444 . . . 4  |-  ( n  =  N  ->  ( dom  ( R ^r 
n )  C_  dom  R  <->  dom  ( R ^r  N )  C_  dom  R ) )
1615imbi2d 314 . . 3  |-  ( n  =  N  ->  (
( R  e.  V  ->  dom  ( R ^r  n )  C_  dom  R )  <->  ( R  e.  V  ->  dom  ( R ^r  N ) 
C_  dom  R )
) )
17 relexp1g 12863 . . . . 5  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
1817dmeqd 5118 . . . 4  |-  ( R  e.  V  ->  dom  ( R ^r 
1 )  =  dom  R )
19 eqimss 3469 . . . 4  |-  ( dom  ( R ^r 
1 )  =  dom  R  ->  dom  ( R ^r  1 ) 
C_  dom  R )
2018, 19syl 16 . . 3  |-  ( R  e.  V  ->  dom  ( R ^r 
1 )  C_  dom  R )
21 relexpsucnnr 12862 . . . . . . . . 9  |-  ( ( R  e.  V  /\  m  e.  NN )  ->  ( R ^r 
( m  +  1 ) )  =  ( ( R ^r 
m )  o.  R
) )
2221ancoms 451 . . . . . . . 8  |-  ( ( m  e.  NN  /\  R  e.  V )  ->  ( R ^r 
( m  +  1 ) )  =  ( ( R ^r 
m )  o.  R
) )
2322dmeqd 5118 . . . . . . 7  |-  ( ( m  e.  NN  /\  R  e.  V )  ->  dom  ( R ^r  ( m  + 
1 ) )  =  dom  ( ( R ^r  m )  o.  R ) )
24 dmcoss 5175 . . . . . . 7  |-  dom  (
( R ^r 
m )  o.  R
)  C_  dom  R
2523, 24syl6eqss 3467 . . . . . 6  |-  ( ( m  e.  NN  /\  R  e.  V )  ->  dom  ( R ^r  ( m  + 
1 ) )  C_  dom  R )
2625a1d 25 . . . . 5  |-  ( ( m  e.  NN  /\  R  e.  V )  ->  ( dom  ( R ^r  m ) 
C_  dom  R  ->  dom  ( R ^r 
( m  +  1 ) )  C_  dom  R ) )
2726ex 432 . . . 4  |-  ( m  e.  NN  ->  ( R  e.  V  ->  ( dom  ( R ^r  m )  C_  dom  R  ->  dom  ( R ^r  ( m  +  1 ) ) 
C_  dom  R )
) )
2827a2d 26 . . 3  |-  ( m  e.  NN  ->  (
( R  e.  V  ->  dom  ( R ^r  m )  C_  dom  R )  ->  ( R  e.  V  ->  dom  ( R ^r 
( m  +  1 ) )  C_  dom  R ) ) )
294, 8, 12, 16, 20, 28nnind 10470 . 2  |-  ( N  e.  NN  ->  ( R  e.  V  ->  dom  ( R ^r  N )  C_  dom  R ) )
3029imp 427 1  |-  ( ( N  e.  NN  /\  R  e.  V )  ->  dom  ( R ^r  N )  C_  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    C_ wss 3389   dom cdm 4913    o. ccom 4917  (class class class)co 6196   1c1 9404    + caddc 9406   NNcn 10452   ^r crelexp 12857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-seq 12011  df-relexp 12858
This theorem is referenced by:  relexpdmg  12877  relexpnnrn  12880  relexpfld  12884  relexpaddg  12888  relexpaddss  38223
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