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Theorem qtopcmplem 19943
Description: Lemma for qtopcmp 19944 and qtopcon 19945. (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 5799 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
42, 3sylib 196 . . 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 19938 . . . . . 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 5791 . . . 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 210 . 2  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  F : X -onto-> U. ( J qTop  F )
)
136toptopon 19201 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
145, 13sylib 196 . . 3  |-  ( J  e.  A  ->  J  e.  (TopOn `  X )
)
15 qtopid 19941 . . 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 1228 1  |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   U.cuni 4245   ran crn 5000    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   qTop cqtop 14754   Topctop 19161  TopOnctopon 19162    Cn ccn 19491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-qtop 14758  df-top 19166  df-topon 19169  df-cn 19494
This theorem is referenced by:  qtopcmp  19944  qtopcon  19945  qtoppcon  28321
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