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Theorem iinon 6903
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 5193 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
21adantr 465 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
3 eqid 2451 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fmpt 5965 . . . . 5  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
5 frn 5665 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  ( x  e.  A  |->  B )  C_  On )
64, 5sylbi 195 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
76adantr 465 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  C_  On )
8 dm0rn0 5156 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
9 dmmptg 5435 . . . . . . 7  |-  ( A. x  e.  A  B  e.  On  ->  dom  ( x  e.  A  |->  B )  =  A )
109eqeq1d 2453 . . . . . 6  |-  ( A. x  e.  A  B  e.  On  ->  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
118, 10syl5bbr 259 . . . . 5  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
1211necon3bid 2706 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =/=  (/)  <->  A  =/=  (/) ) )
1312biimpar 485 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  =/=  (/) )
14 oninton 6513 . . 3  |-  ( ( ran  ( x  e.  A  |->  B )  C_  On  /\  ran  ( x  e.  A  |->  B )  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
157, 13, 14syl2anc 661 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
162, 15eqeltrd 2539 1  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795    C_ wss 3428   (/)c0 3737   |^|cint 4228   |^|_ciin 4272    |-> cmpt 4450   Oncon0 4819   dom cdm 4940   ran crn 4941   -->wf 5514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526
This theorem is referenced by: (None)
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