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Theorem kqfvima 22032
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 6814, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqfvima ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqfvima
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 22027 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1113 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝐹 Fn 𝑋)
4 toponss 21229 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
543adant3 1112 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝑈𝑋)
6 fnfvima 6814 . . . 4 ((𝐹 Fn 𝑋𝑈𝑋𝐴𝑈) → (𝐹𝐴) ∈ (𝐹𝑈))
763expia 1101 . . 3 ((𝐹 Fn 𝑋𝑈𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
83, 5, 7syl2anc 576 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
9 fnfun 6280 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
10 fvelima 6555 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) ∈ (𝐹𝑈)) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴))
1110ex 405 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
123, 9, 113syl 18 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
13 simpl1 1171 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐽 ∈ (TopOn‘𝑋))
145sselda 3854 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑋)
15 simpl3 1173 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐴𝑋)
161kqfeq 22026 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
1713, 14, 15, 16syl3anc 1351 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
18 eleq2 2848 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
19 eleq2 2848 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
2018, 19bibi12d 338 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑤𝐴𝑤)))
2120cbvralv 3377 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤))
2217, 21syl6bb 279 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤)))
23 simpl2 1172 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑈𝐽)
24 eleq2 2848 . . . . . . . . 9 (𝑤 = 𝑈 → (𝑧𝑤𝑧𝑈))
25 eleq2 2848 . . . . . . . . 9 (𝑤 = 𝑈 → (𝐴𝑤𝐴𝑈))
2624, 25bibi12d 338 . . . . . . . 8 (𝑤 = 𝑈 → ((𝑧𝑤𝐴𝑤) ↔ (𝑧𝑈𝐴𝑈)))
2726rspcv 3525 . . . . . . 7 (𝑈𝐽 → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2823, 27syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2922, 28sylbid 232 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → (𝑧𝑈𝐴𝑈)))
30 simpr 477 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑈)
31 biimp 207 . . . . 5 ((𝑧𝑈𝐴𝑈) → (𝑧𝑈𝐴𝑈))
3229, 30, 31syl6ci 71 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3332rexlimdva 3223 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3412, 33syld 47 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → 𝐴𝑈))
358, 34impbid 204 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2048  wral 3082  wrex 3083  {crab 3086  wss 3825  cmpt 5002  cima 5403  Fun wfun 6176   Fn wfn 6177  cfv 6182  TopOnctopon 21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-topon 21213
This theorem is referenced by:  kqsat  22033  kqdisj  22034  kqcldsat  22035  kqt0lem  22038  isr0  22039  regr1lem  22041  kqreglem1  22043  kqreglem2  22044
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