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Theorem qtopcmplem 17398
Description: Lemma for qtopcmp 17399 and qtopcon 17400. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopcmp.1  |-  X  = 
U. J
qtopcmplem.1  |-  ( J  e.  A  ->  J  e.  Top )
qtopcmplem.2  |-  ( ( J  e.  A  /\  F : X -onto-> U. ( J qTop  F )  /\  F  e.  ( J  Cn  ( J qTop  F ) ) )  ->  ( J qTop  F
)  e.  A )
Assertion
Ref Expression
qtopcmplem  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )

Proof of Theorem qtopcmplem
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  J  e.  A )
2 simpr 447 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F  Fn  X )
3 dffn4 5457 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
42, 3sylib 188 . . 3  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F : X -onto-> ran  F )
5 qtopcmplem.1 . . . . . 6  |-  ( J  e.  A  ->  J  e.  Top )
6 qtopcmp.1 . . . . . . 7  |-  X  = 
U. J
76qtopuni 17393 . . . . . 6  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
85, 7sylan 457 . . . . 5  |-  ( ( J  e.  A  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
93, 8sylan2b 461 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ran  F  =  U. ( J qTop  F )
)
10 foeq3 5449 . . . 4  |-  ( ran 
F  =  U. ( J qTop  F )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F ) ) )
119, 10syl 15 . . 3  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F )
) )
124, 11mpbid 201 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F : X -onto-> U. ( J qTop  F )
)
136toptopon 16671 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
145, 13sylib 188 . . 3  |-  ( J  e.  A  ->  J  e.  (TopOn `  X )
)
15 qtopid 17396 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
1614, 15sylan 457 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F
) ) )
17 qtopcmplem.2 . 2  |-  ( ( J  e.  A  /\  F : X -onto-> U. ( J qTop  F )  /\  F  e.  ( J  Cn  ( J qTop  F ) ) )  ->  ( J qTop  F
)  e.  A )
181, 12, 16, 17syl3anc 1182 1  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   U.cuni 3827   ran crn 4690    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  qtopcmp  17399  qtopcon  17400  qtoppcon  23767
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cn 16957
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