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Theorem iinon 6947
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 5182 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
21adantr 463 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
3 eqid 2392 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43rnmptss 5975 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
54adantr 463 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  C_  On )
6 dm0rn0 5145 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
7 dmmptg 5425 . . . . . . 7  |-  ( A. x  e.  A  B  e.  On  ->  dom  ( x  e.  A  |->  B )  =  A )
87eqeq1d 2394 . . . . . 6  |-  ( A. x  e.  A  B  e.  On  ->  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
96, 8syl5bbr 259 . . . . 5  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  A  =  (/) ) )
109necon3bid 2650 . . . 4  |-  ( A. x  e.  A  B  e.  On  ->  ( ran  ( x  e.  A  |->  B )  =/=  (/)  <->  A  =/=  (/) ) )
1110biimpar 483 . . 3  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  ran  ( x  e.  A  |->  B )  =/=  (/) )
12 oninton 6552 . . 3  |-  ( ( ran  ( x  e.  A  |->  B )  C_  On  /\  ran  ( x  e.  A  |->  B )  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
135, 11, 12syl2anc 659 . 2  |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  On )
142, 13eqeltrd 2480 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 367    = wceq 1399    e. wcel 1836    =/= wne 2587   A.wral 2742    C_ wss 3402   (/)c0 3724   |^|cint 4212   |^|_ciin 4257    |-> cmpt 4438   Oncon0 4805   dom cdm 4926   ran crn 4927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iin 4259  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fv 5517
This theorem is referenced by: (None)
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