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Theorem qtoptop 20385
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 455 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  J  e.  Top )
2 id 22 . . 3  |-  ( F  Fn  X  ->  F  Fn  X )
3 qtoptop.1 . . . 4  |-  X  = 
U. J
43topopn 19599 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
5 fnex 6076 . . 3  |-  ( ( F  Fn  X  /\  X  e.  J )  ->  F  e.  _V )
62, 4, 5syl2anr 476 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  F  e.  _V )
7 fnfun 5615 . . 3  |-  ( F  Fn  X  ->  Fun  F )
87adantl 464 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  Fun  F )
9 qtoptop2 20384 . 2  |-  ( ( J  e.  Top  /\  F  e.  _V  /\  Fun  F )  ->  ( J qTop  F )  e.  Top )
101, 6, 8, 9syl3anc 1230 1  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   U.cuni 4190   Fun wfun 5519    Fn wfn 5520  (class class class)co 6234   qTop cqtop 15009   Topctop 19578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-qtop 15013  df-top 19583
This theorem is referenced by:  qtoptopon  20389  qtopkgen  20395  qtopt1  28171  qtophaus  28172
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