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Theorem 1stcrestlem 17194
Description: Lemma for 1stcrest 17195. (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 4681 . . . . . 6  |-  Ord  om
2 reldom 6885 . . . . . . . 8  |-  Rel  ~<_
32brrelex2i 4746 . . . . . . 7  |-  ( B  ~<_  om  ->  om  e.  _V )
4 elong 4416 . . . . . . 7  |-  ( om  e.  _V  ->  ( om  e.  On  <->  Ord  om )
)
53, 4syl 15 . . . . . 6  |-  ( B  ~<_  om  ->  ( om  e.  On  <->  Ord  om ) )
61, 5mpbiri 224 . . . . 5  |-  ( B  ~<_  om  ->  om  e.  On )
7 ondomen 7680 . . . . 5  |-  ( ( om  e.  On  /\  B  ~<_  om )  ->  B  e.  dom  card )
86, 7mpancom 650 . . . 4  |-  ( B  ~<_  om  ->  B  e.  dom  card )
9 eqid 2296 . . . . 5  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
109dmmptss 5185 . . . 4  |-  dom  (
x  e.  B  |->  C )  C_  B
11 ssnum 7682 . . . 4  |-  ( ( B  e.  dom  card  /\ 
dom  ( x  e.  B  |->  C )  C_  B )  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
128, 10, 11sylancl 643 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
13 funmpt 5306 . . . 4  |-  Fun  (
x  e.  B  |->  C )
14 funforn 5474 . . . 4  |-  ( Fun  ( x  e.  B  |->  C )  <->  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C ) )
1513, 14mpbi 199 . . 3  |-  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C )
16 fodomnum 7700 . . 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 1366 . 2  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  dom  ( x  e.  B  |->  C ) )
182brrelexi 4745 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
19 ssdomg 6923 . . . 4  |-  ( B  e.  _V  ->  ( dom  ( x  e.  B  |->  C )  C_  B  ->  dom  ( x  e.  B  |->  C )  ~<_  B ) )
2018, 10, 19ee10 1366 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  B )
21 domtr 6930 . . 3  |-  ( ( dom  ( x  e.  B  |->  C )  ~<_  B  /\  B  ~<_  om )  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
2220, 21mpancom 650 . 2  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
23 domtr 6930 . 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 642 1  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   Ord word 4407   Oncon0 4408   omcom 4672   dom cdm 4705   ran crn 4706   Fun wfun 5265   -onto->wfo 5269    ~<_ cdom 6877   cardccrd 7584
This theorem is referenced by:  1stcrest  17195  2ndcrest  17196  lly1stc  17238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-card 7588  df-acn 7591
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