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Theorem qtopcn 22322
 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 5920 . . . . . . 7 (𝐺𝑥) ⊆ dom 𝐺
2 simplrr 777 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐺:𝑌𝑍)
31, 2fssdm 6508 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → (𝐺𝑥) ⊆ 𝑌)
4 simplll 774 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 simplrl 776 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → 𝐹:𝑋onto𝑌)
6 elqtop3 22311 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
74, 5, 6syl2anc 587 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝑥) ⊆ 𝑌 ∧ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)))
83, 7mpbirand 706 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽))
9 cnvco 5724 . . . . . . . 8 (𝐺𝐹) = (𝐹𝐺)
109imaeq1i 5897 . . . . . . 7 ((𝐺𝐹) “ 𝑥) = ((𝐹𝐺) “ 𝑥)
11 imaco 6075 . . . . . . 7 ((𝐹𝐺) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1210, 11eqtri 2824 . . . . . 6 ((𝐺𝐹) “ 𝑥) = (𝐹 “ (𝐺𝑥))
1312eleq1i 2883 . . . . 5 (((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ (𝐹 “ (𝐺𝑥)) ∈ 𝐽)
148, 13syl6bbr 292 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) ∧ 𝑥𝐾) → ((𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
1514ralbidva 3164 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽))
16 simprr 772 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐺:𝑌𝑍)
1716biantrurd 536 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐺:𝑌𝑍 ∧ ∀𝑥𝐾 (𝐺𝑥) ∈ (𝐽 qTop 𝐹))))
18 fof 6569 . . . . . 6 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
1918ad2antrl 727 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐹:𝑋𝑌)
20 fco 6509 . . . . 5 ((𝐺:𝑌𝑍𝐹:𝑋𝑌) → (𝐺𝐹):𝑋𝑍)
2116, 19, 20syl2anc 587 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺𝐹):𝑋𝑍)
2221biantrurd 536 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽 ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
2315, 17, 223bitr3d 312 . 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 22312 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2524ad2ant2r 746 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26 simplr 768 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27 iscn 21843 . . 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 587 . 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 21843 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3029adantr 484 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → ((𝐺𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺𝐹):𝑋𝑍 ∧ ∀𝑥𝐾 ((𝐺𝐹) “ 𝑥) ∈ 𝐽)))
3123, 28, 303bitr4d 314 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 209   ∧ wa 399   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ◡ccnv 5522   “ cima 5526   ∘ ccom 5527  ⟶wf 6324  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7139   qTop cqtop 16771  TopOnctopon 21518   Cn ccn 21832 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-qtop 16775  df-top 21502  df-topon 21519  df-cn 21835 This theorem is referenced by:  qtopeu  22324
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