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Theorem qtopcmplem 20502
Description: Lemma for qtopcmp 20503 and qtopcon 20504. (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 457 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  J  e.  A )
2 simpr 461 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F  Fn  X )
3 dffn4 5786 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
42, 3sylib 198 . . 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 20497 . . . . . 6  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
85, 7sylan 471 . . . . 5  |-  ( ( J  e.  A  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
93, 8sylan2b 475 . . . 4  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ran  F  =  U. ( J qTop  F )
)
10 foeq3 5778 . . . 4  |-  ( ran 
F  =  U. ( J qTop  F )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F ) ) )
119, 10syl 17 . . 3  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( F : X -onto-> ran  F  <->  F : X -onto-> U. ( J qTop  F )
) )
124, 11mpbid 212 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F : X -onto-> U. ( J qTop  F )
)
136toptopon 19728 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
145, 13sylib 198 . . 3  |-  ( J  e.  A  ->  J  e.  (TopOn `  X )
)
15 qtopid 20500 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
1614, 15sylan 471 . 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 1232 1  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   U.cuni 4193   ran crn 4826    Fn wfn 5566   -onto->wfo 5569   ` cfv 5571  (class class class)co 6280   qTop cqtop 15119   Topctop 19688  TopOnctopon 19689    Cn ccn 20020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-qtop 15123  df-top 19693  df-topon 19696  df-cn 20023
This theorem is referenced by:  qtopcmp  20503  qtopcon  20504  qtoppcon  29546
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