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Theorem qtopcmplem 17414
Description: Lemma for qtopcmp 17415 and qtopcon 17416. (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 5473 . . . 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 17409 . . . . . 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 5465 . . . 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 16687 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
145, 13sylib 188 . . 3  |-  ( J  e.  A  ->  J  e.  (TopOn `  X )
)
15 qtopid 17412 . . 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 1632    e. wcel 1696   U.cuni 3843   ran crn 4706    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   qTop cqtop 13422   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  qtopcmp  17415  qtopcon  17416  qtoppcon  23782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cn 16973
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