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Theorem qtoptop 17608
Description: The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtoptop.1  |-  X  = 
U. J
Assertion
Ref Expression
qtoptop  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )

Proof of Theorem qtoptop
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  J  e.  Top )
2 id 19 . . 3  |-  ( F  Fn  X  ->  F  Fn  X )
3 qtoptop.1 . . . 4  |-  X  = 
U. J
43topopn 16869 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
5 fnex 5861 . . 3  |-  ( ( F  Fn  X  /\  X  e.  J )  ->  F  e.  _V )
62, 4, 5syl2anr 464 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  F  e.  _V )
7 fnfun 5446 . . 3  |-  ( F  Fn  X  ->  Fun  F )
87adantl 452 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  Fun  F )
9 qtoptop2 17607 . 2  |-  ( ( J  e.  Top  /\  F  e.  _V  /\  Fun  F )  ->  ( J qTop  F )  e.  Top )
101, 6, 8, 9syl3anc 1183 1  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   U.cuni 3929   Fun wfun 5352    Fn wfn 5353  (class class class)co 5981   qTop cqtop 13616   Topctop 16848
This theorem is referenced by:  qtoptopon  17612  qtopkgen  17618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-qtop 13620  df-top 16853
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