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Theorem 1stcrestlem 17468
Description: Lemma for 1stcrest 17469. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
1stcrestlem  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
Distinct variable group:    x, B
Allowed substitution hint:    C( x)

Proof of Theorem 1stcrestlem
StepHypRef Expression
1 ordom 4813 . . . . . 6  |-  Ord  om
2 reldom 7074 . . . . . . . 8  |-  Rel  ~<_
32brrelex2i 4878 . . . . . . 7  |-  ( B  ~<_  om  ->  om  e.  _V )
4 elong 4549 . . . . . . 7  |-  ( om  e.  _V  ->  ( om  e.  On  <->  Ord  om )
)
53, 4syl 16 . . . . . 6  |-  ( B  ~<_  om  ->  ( om  e.  On  <->  Ord  om ) )
61, 5mpbiri 225 . . . . 5  |-  ( B  ~<_  om  ->  om  e.  On )
7 ondomen 7874 . . . . 5  |-  ( ( om  e.  On  /\  B  ~<_  om )  ->  B  e.  dom  card )
86, 7mpancom 651 . . . 4  |-  ( B  ~<_  om  ->  B  e.  dom  card )
9 eqid 2404 . . . . 5  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
109dmmptss 5325 . . . 4  |-  dom  (
x  e.  B  |->  C )  C_  B
11 ssnum 7876 . . . 4  |-  ( ( B  e.  dom  card  /\ 
dom  ( x  e.  B  |->  C )  C_  B )  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
128, 10, 11sylancl 644 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
13 funmpt 5448 . . . 4  |-  Fun  (
x  e.  B  |->  C )
14 funforn 5619 . . . 4  |-  ( Fun  ( x  e.  B  |->  C )  <->  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C ) )
1513, 14mpbi 200 . . 3  |-  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C )
16 fodomnum 7894 . . 3  |-  ( dom  ( x  e.  B  |->  C )  e.  dom  card 
->  ( ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C ) -onto-> ran  ( x  e.  B  |->  C )  ->  ran  ( x  e.  B  |->  C )  ~<_  dom  (
x  e.  B  |->  C ) ) )
1712, 15, 16ee10 1382 . 2  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  dom  ( x  e.  B  |->  C ) )
182brrelexi 4877 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
19 ssdomg 7112 . . . 4  |-  ( B  e.  _V  ->  ( dom  ( x  e.  B  |->  C )  C_  B  ->  dom  ( x  e.  B  |->  C )  ~<_  B ) )
2018, 10, 19ee10 1382 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  B )
21 domtr 7119 . . 3  |-  ( ( dom  ( x  e.  B  |->  C )  ~<_  B  /\  B  ~<_  om )  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
2220, 21mpancom 651 . 2  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
23 domtr 7119 . 2  |-  ( ( ran  ( x  e.  B  |->  C )  ~<_  dom  ( x  e.  B  |->  C )  /\  dom  ( x  e.  B  |->  C )  ~<_  om )  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
2417, 22, 23syl2anc 643 1  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   _Vcvv 2916    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   Ord word 4540   Oncon0 4541   omcom 4804   dom cdm 4837   ran crn 4838   Fun wfun 5407   -onto->wfo 5411    ~<_ cdom 7066   cardccrd 7778
This theorem is referenced by:  1stcrest  17469  2ndcrest  17470  lly1stc  17512  abrexct  24064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-card 7782  df-acn 7785
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