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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  ontowfo 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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