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Theorem iinon 6357
Description: The nonempty indexed intersection of a class of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iinon  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  On )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem iinon
StepHypRef Expression
1 dfiin3g 4932 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
21adantr 451 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
3 eqid 2283 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fmpt 5681 . . . . 5  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
5 frn 5395 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  ( x  e.  A  |->  B )  C_  On )
64, 5sylbi 187 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
76adantr 451 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  C_  On )
8 dm0rn0 4895 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
9 dmmptg 5170 . . . . . . 7  |-  ( A. x  e.  A  B  e.  On  ->  dom  ( x  e.  A  |->  B )  =  A )
109eqeq1d 2291 . . . . . 6  |-  ( A. x  e.  A  B  e.  On  ->  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
118, 10syl5bbr 250 . . . . 5  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
1211necon3bid 2481 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =/=  (/)  <->  A  =/=  (/) ) )
1312biimpar 471 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  =/=  (/) )
14 oninton 4591 . . 3  |-  ( ( ran  ( x  e.  A  |->  B )  C_  On  /\  ran  ( x  e.  A  |->  B )  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
157, 13, 14syl2anc 642 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
162, 15eqeltrd 2357 1  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|cint 3862   |^|_ciin 3906    e. cmpt 4077   Oncon0 4392   dom cdm 4689   ran crn 4690   -->wf 5251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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