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Theorem qtoptop 19398
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 457 . 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 18644 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
5 fnex 6046 . . 3  |-  ( ( F  Fn  X  /\  X  e.  J )  ->  F  e.  _V )
62, 4, 5syl2anr 478 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  F  e.  _V )
7 fnfun 5609 . . 3  |-  ( F  Fn  X  ->  Fun  F )
87adantl 466 . 2  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  Fun  F )
9 qtoptop2 19397 . 2  |-  ( ( J  e.  Top  /\  F  e.  _V  /\  Fun  F )  ->  ( J qTop  F )  e.  Top )
101, 6, 8, 9syl3anc 1219 1  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   U.cuni 4192   Fun wfun 5513    Fn wfn 5514  (class class class)co 6193   qTop cqtop 14552   Topctop 18623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-qtop 14556  df-top 18628
This theorem is referenced by:  qtoptopon  19402  qtopkgen  19408
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