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Theorem iinon 6373
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 4948 . . 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 2296 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fmpt 5697 . . . . 5  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
5 frn 5411 . . . . 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 4911 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
9 dmmptg 5186 . . . . . . 7  |-  ( A. x  e.  A  B  e.  On  ->  dom  ( x  e.  A  |->  B )  =  A )
109eqeq1d 2304 . . . . . 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 2494 . . . 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 4607 . . 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 2370 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   |^|cint 3878   |^|_ciin 3922    e. cmpt 4093   Oncon0 4408   dom cdm 4705   ran crn 4706   -->wf 5267
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-sep 4157  ax-nul 4165  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iin 3924  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-we 4370  df-ord 4411  df-on 4412  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-fv 5279
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