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Theorem kqfvima 23738
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7253, 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 23733 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1134 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝐹 Fn 𝑋)
4 toponss 22933 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
543adant3 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝑈𝑋)
6 fnfvima 7253 . . . 4 ((𝐹 Fn 𝑋𝑈𝑋𝐴𝑈) → (𝐹𝐴) ∈ (𝐹𝑈))
763expia 1122 . . 3 ((𝐹 Fn 𝑋𝑈𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
83, 5, 7syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
9 fnfun 6668 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
10 fvelima 6974 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) ∈ (𝐹𝑈)) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴))
1110ex 412 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
123, 9, 113syl 18 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
13 simpl1 1192 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐽 ∈ (TopOn‘𝑋))
145sselda 3983 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑋)
15 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐴𝑋)
161kqfeq 23732 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
1713, 14, 15, 16syl3anc 1373 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
18 eleq2 2830 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
19 eleq2 2830 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
2018, 19bibi12d 345 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑤𝐴𝑤)))
2120cbvralvw 3237 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤))
2217, 21bitrdi 287 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤)))
23 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑈𝐽)
24 eleq2 2830 . . . . . . . . 9 (𝑤 = 𝑈 → (𝑧𝑤𝑧𝑈))
25 eleq2 2830 . . . . . . . . 9 (𝑤 = 𝑈 → (𝐴𝑤𝐴𝑈))
2624, 25bibi12d 345 . . . . . . . 8 (𝑤 = 𝑈 → ((𝑧𝑤𝐴𝑤) ↔ (𝑧𝑈𝐴𝑈)))
2726rspcv 3618 . . . . . . 7 (𝑈𝐽 → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2823, 27syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2922, 28sylbid 240 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → (𝑧𝑈𝐴𝑈)))
30 simpr 484 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑈)
31 biimp 215 . . . . 5 ((𝑧𝑈𝐴𝑈) → (𝑧𝑈𝐴𝑈))
3229, 30, 31syl6ci 71 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3332rexlimdva 3155 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3412, 33syld 47 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → 𝐴𝑈))
358, 34impbid 212 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951  cmpt 5225  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561  TopOnctopon 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-topon 22917
This theorem is referenced by:  kqsat  23739  kqdisj  23740  kqcldsat  23741  kqt0lem  23744  isr0  23745  regr1lem  23747  kqreglem1  23749  kqreglem2  23750
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