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Theorem qtopcn 21327
 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 simplll 794 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 simplrl 796 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐹:𝑋onto𝑌)
3 elqtop3 21316 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
41, 2, 3syl2anc 691 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
5 cnvimass 5404 . . . . . . . 8 (𝐺𝑥) ⊆ dom 𝐺
6 simplrr 797 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐺:𝑌𝑍)
7 fdm 5964 . . . . . . . . 9 (𝐺:𝑌𝑍 → dom 𝐺 = 𝑌)
86, 7syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → dom 𝐺 = 𝑌)
95, 8syl5sseq 3616 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → (𝐺𝑥) ⊆ 𝑌)
109biantrurd 528 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐹 “ (𝐺𝑥)) ∈ 𝐽 ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
114, 10bitr4d 270 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽))
12 cnvco 5230 . . . . . . . 8 (𝐺𝐹) = (𝐹𝐺)
1312imaeq1i 5382 . . . . . . 7 ((𝐺𝐹) “ 𝑥) = ((𝐹𝐺) “ 𝑥)
14 imaco 5557 . . . . . . 7 ((𝐹𝐺) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1513, 14eqtri 2632 . . . . . 6 ((𝐺𝐹) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1615eleq1i 2679 . . . . 5 (((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)
1711, 16syl6bbr 277 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
1817ralbidva 2968 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
19 simprr 792 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐺:𝑌𝑍)
2019biantrurd 528 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
21 fof 6028 . . . . . 6 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
2221ad2antrl 760 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐹:𝑋𝑌)
23 fco 5971 . . . . 5 ((𝐺:𝑌𝑍𝐹:𝑋𝑌) → (𝐺𝐹):𝑋𝑍)
2419, 22, 23syl2anc 691 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺𝐹):𝑋𝑍)
2524biantrurd 528 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
2618, 20, 253bitr3d 297 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹)) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
27 qtoptopon 21317 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2827ad2ant2r 779 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
29 simplr 788 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
30 iscn 20849 . . 3 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
3128, 29, 30syl2anc 691 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
32 iscn 20849 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3332adantr 480 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3426, 31, 333bitr4d 299 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ◡ccnv 5037  dom cdm 5038   “ cima 5041   ∘ ccom 5042  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  TopOnctopon 20518   Cn ccn 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cn 20841 This theorem is referenced by:  qtopeu  21329
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