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Theorem onssr1 8310
Description: Initial segments of the ordinals are contained in initial segments of the cumulative hierarchy. (Contributed by FL, 20-Apr-2011.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
onssr1  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )

Proof of Theorem onssr1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 r1funlim 8245 . . . . . . . . . 10  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 463 . . . . . . . . 9  |-  Lim  dom  R1
3 limord 5501 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordtr1 5485 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 ) )
52, 3, 4mp2b 10 . . . . . . . 8  |-  ( ( x  e.  A  /\  A  e.  dom  R1 )  ->  x  e.  dom  R1 )
65ancoms 454 . . . . . . 7  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  dom  R1 )
7 rankonidlem 8307 . . . . . . 7  |-  ( x  e.  dom  R1  ->  ( x  e.  U. ( R1 " On )  /\  ( rank `  x )  =  x ) )
86, 7syl 17 . . . . . 6  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e. 
U. ( R1 " On )  /\  ( rank `  x )  =  x ) )
98simprd 464 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( rank `  x
)  =  x )
10 simpr 462 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  A
)
119, 10eqeltrd 2507 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( rank `  x
)  e.  A )
128simpld 460 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
13 simpl 458 . . . . 5  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  A  e.  dom  R1 )
14 rankr1ag 8281 . . . . 5  |-  ( ( x  e.  U. ( R1 " On )  /\  A  e.  dom  R1 )  ->  ( x  e.  ( R1 `  A
)  <->  ( rank `  x
)  e.  A ) )
1512, 13, 14syl2anc 665 . . . 4  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  ( x  e.  ( R1 `  A
)  <->  ( rank `  x
)  e.  A ) )
1611, 15mpbird 235 . . 3  |-  ( ( A  e.  dom  R1  /\  x  e.  A )  ->  x  e.  ( R1 `  A ) )
1716ex 435 . 2  |-  ( A  e.  dom  R1  ->  ( x  e.  A  ->  x  e.  ( R1 `  A ) ) )
1817ssrdv 3470 1  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   U.cuni 4219   dom cdm 4853   "cima 4856   Ord word 5441   Oncon0 5442   Lim wlim 5443   Fun wfun 5595   ` cfv 5601   R1cr1 8241   rankcrnk 8242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243  df-rank 8244
This theorem is referenced by:  rankr1id  8341  ackbij2  8680  wunom  9152  r1limwun  9168  inar1  9207  r1tskina  9214
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