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Theorem qtopres 17735
 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 5181 . . . . . . 7
21pweqi 3805 . . . . . 6
3 rabeq 2952 . . . . . 6
42, 3ax-mp 5 . . . . 5
5 residm 5180 . . . . . . . . . . 11
65cnveqi 5050 . . . . . . . . . 10
76imaeq1i 5203 . . . . . . . . 9
8 cnvresima 5362 . . . . . . . . 9
9 cnvresima 5362 . . . . . . . . 9
107, 8, 93eqtr3i 2466 . . . . . . . 8
1110eleq1i 2501 . . . . . . 7
1211a1i 11 . . . . . 6
1312rabbiia 2948 . . . . 5
144, 13eqtr2i 2459 . . . 4
15 qtopval.1 . . . . 5
1615qtopval 17732 . . . 4 qTop
17 resexg 5188 . . . . 5
1815qtopval 17732 . . . . 5 qTop
1917, 18sylan2 462 . . . 4 qTop
2014, 16, 193eqtr4a 2496 . . 3 qTop qTop
2120expcom 426 . 2 qTop qTop
22 df-qtop 13738 . . . . 5 qTop
2322reldmmpt2 6184 . . . 4 qTop
2423ovprc1 6112 . . 3 qTop
2523ovprc1 6112 . . 3 qTop
2624, 25eqtr4d 2473 . 2 qTop qTop
2721, 26pm2.61d1 154 1 qTop qTop
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  crab 2711  cvv 2958   cin 3321  c0 3630  cpw 3801  cuni 4017  ccnv 4880   cres 4883  cima 4884  (class class class)co 6084   qTop cqtop 13734 This theorem is referenced by:  qtoptop2  17736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-qtop 13738
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