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Theorem qtopcn 23209
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 6077 . . . . . . 7 (𝐺𝑥) ⊆ dom 𝐺
2 simplrr 776 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐺:𝑌𝑍)
31, 2fssdm 6734 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → (𝐺𝑥) ⊆ 𝑌)
4 simplll 773 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 simplrl 775 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐹:𝑋onto𝑌)
6 elqtop3 23198 . . . . . . 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 705 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽))
9 cnvco 5883 . . . . . . . 8 (𝐺𝐹) = (𝐹𝐺)
109imaeq1i 6054 . . . . . . 7 ((𝐺𝐹) “ 𝑥) = ((𝐹𝐺) “ 𝑥)
11 imaco 6247 . . . . . . 7 ((𝐹𝐺) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1210, 11eqtri 2760 . . . . . 6 ((𝐺𝐹) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1312eleq1i 2824 . . . . 5 (((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)
148, 13bitr4di 288 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
1514ralbidva 3175 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
16 simprr 771 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐺:𝑌𝑍)
1716biantrurd 533 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
18 fof 6802 . . . . . 6 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
1918ad2antrl 726 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐹:𝑋𝑌)
20 fco 6738 . . . . 5 ((𝐺:𝑌𝑍𝐹:𝑋𝑌) → (𝐺𝐹):𝑋𝑍)
2116, 19, 20syl2anc 584 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺𝐹):𝑋𝑍)
2221biantrurd 533 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
2315, 17, 223bitr3d 308 . 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 23199 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2524ad2ant2r 745 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26 simplr 767 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27 iscn 22730 . . 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 22730 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3029adantr 481 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3123, 28, 303bitr4d 310 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 205  wa 396  wcel 2106  wral 3061  wss 3947  ccnv 5674  cima 5678  ccom 5679  wf 6536  ontowfo 6538  cfv 6540  (class class class)co 7405   qTop cqtop 17445  TopOnctopon 22403   Cn ccn 22719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-qtop 17449  df-top 22387  df-topon 22404  df-cn 22722
This theorem is referenced by:  qtopeu  23211
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