MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qtopcn Structured version   Visualization version   GIF version

Theorem qtopcn 23738
Description: Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtopcn (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺𝐹) ∈ (𝐽 Cn 𝐾)))

Proof of Theorem qtopcn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cnvimass 6102 . . . . . . 7 (𝐺𝑥) ⊆ dom 𝐺
2 simplrr 778 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐺:𝑌𝑍)
31, 2fssdm 6756 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → (𝐺𝑥) ⊆ 𝑌)
4 simplll 775 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 simplrl 777 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐹:𝑋onto𝑌)
6 elqtop3 23727 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
74, 5, 6syl2anc 584 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
83, 7mpbirand 707 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽))
9 cnvco 5899 . . . . . . . 8 (𝐺𝐹) = (𝐹𝐺)
109imaeq1i 6077 . . . . . . 7 ((𝐺𝐹) “ 𝑥) = ((𝐹𝐺) “ 𝑥)
11 imaco 6273 . . . . . . 7 ((𝐹𝐺) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1210, 11eqtri 2763 . . . . . 6 ((𝐺𝐹) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1312eleq1i 2830 . . . . 5 (((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)
148, 13bitr4di 289 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
1514ralbidva 3174 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
16 simprr 773 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐺:𝑌𝑍)
1716biantrurd 532 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
18 fof 6821 . . . . . 6 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
1918ad2antrl 728 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐹:𝑋𝑌)
20 fco 6761 . . . . 5 ((𝐺:𝑌𝑍𝐹:𝑋𝑌) → (𝐺𝐹):𝑋𝑍)
2116, 19, 20syl2anc 584 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺𝐹):𝑋𝑍)
2221biantrurd 532 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
2315, 17, 223bitr3d 309 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹)) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
24 qtoptopon 23728 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2524ad2ant2r 747 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26 simplr 769 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27 iscn 23259 . . 3 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
2825, 26, 27syl2anc 584 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
29 iscn 23259 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3029adantr 480 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3123, 28, 303bitr4d 311 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺𝐹) ∈ (𝐽 Cn 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wral 3059  wss 3963  ccnv 5688  cima 5692  ccom 5693  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431   qTop cqtop 17550  TopOnctopon 22932   Cn ccn 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-qtop 17554  df-top 22916  df-topon 22933  df-cn 23251
This theorem is referenced by:  qtopeu  23740
  Copyright terms: Public domain W3C validator