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Theorem 1stcrestlem 17517
Description: Lemma for 1stcrest 17518. (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 4856 . . . . . 6  |-  Ord  om
2 reldom 7117 . . . . . . . 8  |-  Rel  ~<_
32brrelex2i 4921 . . . . . . 7  |-  ( B  ~<_  om  ->  om  e.  _V )
4 elong 4591 . . . . . . 7  |-  ( om  e.  _V  ->  ( om  e.  On  <->  Ord  om )
)
53, 4syl 16 . . . . . 6  |-  ( B  ~<_  om  ->  ( om  e.  On  <->  Ord  om ) )
61, 5mpbiri 226 . . . . 5  |-  ( B  ~<_  om  ->  om  e.  On )
7 ondomen 7920 . . . . 5  |-  ( ( om  e.  On  /\  B  ~<_  om )  ->  B  e.  dom  card )
86, 7mpancom 652 . . . 4  |-  ( B  ~<_  om  ->  B  e.  dom  card )
9 eqid 2438 . . . . 5  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
109dmmptss 5368 . . . 4  |-  dom  (
x  e.  B  |->  C )  C_  B
11 ssnum 7922 . . . 4  |-  ( ( B  e.  dom  card  /\ 
dom  ( x  e.  B  |->  C )  C_  B )  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
128, 10, 11sylancl 645 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  e.  dom  card )
13 funmpt 5491 . . . 4  |-  Fun  (
x  e.  B  |->  C )
14 funforn 5662 . . . 4  |-  ( Fun  ( x  e.  B  |->  C )  <->  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C ) )
1513, 14mpbi 201 . . 3  |-  ( x  e.  B  |->  C ) : dom  ( x  e.  B  |->  C )
-onto->
ran  ( x  e.  B  |->  C )
16 fodomnum 7940 . . 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 1386 . 2  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  dom  ( x  e.  B  |->  C ) )
182brrelexi 4920 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
19 ssdomg 7155 . . . 4  |-  ( B  e.  _V  ->  ( dom  ( x  e.  B  |->  C )  C_  B  ->  dom  ( x  e.  B  |->  C )  ~<_  B ) )
2018, 10, 19ee10 1386 . . 3  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  B )
21 domtr 7162 . . 3  |-  ( ( dom  ( x  e.  B  |->  C )  ~<_  B  /\  B  ~<_  om )  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
2220, 21mpancom 652 . 2  |-  ( B  ~<_  om  ->  dom  ( x  e.  B  |->  C )  ~<_  om )
23 domtr 7162 . 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 644 1  |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   _Vcvv 2958    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   Ord word 4582   Oncon0 4583   omcom 4847   dom cdm 4880   ran crn 4881   Fun wfun 5450   -onto->wfo 5454    ~<_ cdom 7109   cardccrd 7824
This theorem is referenced by:  1stcrest  17518  2ndcrest  17519  lly1stc  17561  abrexct  24113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-card 7828  df-acn 7831
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