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Theorem iinon 6602
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 5123 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
21adantr 452 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
3 eqid 2436 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fmpt 5890 . . . . 5  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
5 frn 5597 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  ( x  e.  A  |->  B )  C_  On )
64, 5sylbi 188 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
76adantr 452 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  C_  On )
8 dm0rn0 5086 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
9 dmmptg 5367 . . . . . . 7  |-  ( A. x  e.  A  B  e.  On  ->  dom  ( x  e.  A  |->  B )  =  A )
109eqeq1d 2444 . . . . . 6  |-  ( A. x  e.  A  B  e.  On  ->  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
118, 10syl5bbr 251 . . . . 5  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
1211necon3bid 2636 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =/=  (/)  <->  A  =/=  (/) ) )
1312biimpar 472 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  =/=  (/) )
14 oninton 4780 . . 3  |-  ( ( ran  ( x  e.  A  |->  B )  C_  On  /\  ran  ( x  e.  A  |->  B )  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
157, 13, 14syl2anc 643 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
162, 15eqeltrd 2510 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    C_ wss 3320   (/)c0 3628   |^|cint 4050   |^|_ciin 4094    e. cmpt 4266   Oncon0 4581   dom cdm 4878   ran crn 4879   -->wf 5450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462
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