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Theorem qtopcmplem 17741
Description: Lemma for qtopcmp 17742 and qtopcon 17743. (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 445 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  J  e.  A )
2 simpr 449 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F  Fn  X )
3 dffn4 5661 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
42, 3sylib 190 . . 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 17736 . . . . . 6  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
85, 7sylan 459 . . . . 5  |-  ( ( J  e.  A  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
93, 8sylan2b 463 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ran  F  =  U. ( J qTop  F )
)
10 foeq3 5653 . . . 4  |-  ( ran 
F  =  U. ( J qTop  F )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F ) ) )
119, 10syl 16 . . 3  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F )
) )
124, 11mpbid 203 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F : X -onto-> U. ( J qTop  F )
)
136toptopon 17000 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
145, 13sylib 190 . . 3  |-  ( J  e.  A  ->  J  e.  (TopOn `  X )
)
15 qtopid 17739 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
1614, 15sylan 459 . 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 1185 1  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   U.cuni 4017   ran crn 4881    Fn wfn 5451   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   qTop cqtop 13731   Topctop 16960  TopOnctopon 16961    Cn ccn 17290
This theorem is referenced by:  qtopcmp  17742  qtopcon  17743  qtoppcon  24925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-qtop 13735  df-top 16965  df-topon 16968  df-cn 17293
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