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Theorem qtopres 20790
 Description: The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that be a function with domain . (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1
Assertion
Ref Expression
qtopres qTop qTop

Proof of Theorem qtopres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resima 5143 . . . . . . 7
21pweqi 3946 . . . . . 6
3 rabeq 3024 . . . . . 6
42, 3ax-mp 5 . . . . 5
5 residm 5142 . . . . . . . . . . 11
65cnveqi 5014 . . . . . . . . . 10
76imaeq1i 5171 . . . . . . . . 9
8 cnvresima 5331 . . . . . . . . 9
9 cnvresima 5331 . . . . . . . . 9
107, 8, 93eqtr3i 2501 . . . . . . . 8
1110eleq1i 2540 . . . . . . 7
1211a1i 11 . . . . . 6
1312rabbiia 3019 . . . . 5
144, 13eqtr2i 2494 . . . 4
15 qtopval.1 . . . . 5
1615qtopval 20787 . . . 4 qTop
17 resexg 5153 . . . . 5
1815qtopval 20787 . . . . 5 qTop
1917, 18sylan2 482 . . . 4 qTop
2014, 16, 193eqtr4a 2531 . . 3 qTop qTop
2120expcom 442 . 2 qTop qTop
22 df-qtop 15484 . . . . 5 qTop
2322reldmmpt2 6426 . . . 4 qTop
2423ovprc1 6339 . . 3 qTop
2523ovprc1 6339 . . 3 qTop
2624, 25eqtr4d 2508 . 2 qTop qTop
2721, 26pm2.61d1 164 1 qTop qTop
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  crab 2760  cvv 3031   cin 3389  c0 3722  cpw 3942  cuni 4190  ccnv 4838   cres 4841  cima 4842  (class class class)co 6308   qTop cqtop 15479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-qtop 15484 This theorem is referenced by:  qtoptop2  20791
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