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Theorem kqfvima 21343
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 6400, 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 21338 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1075 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝐹 Fn 𝑋)
4 toponss 20544 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
543adant3 1074 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝑈𝑋)
6 fnfvima 6400 . . . 4 ((𝐹 Fn 𝑋𝑈𝑋𝐴𝑈) → (𝐹𝐴) ∈ (𝐹𝑈))
763expia 1259 . . 3 ((𝐹 Fn 𝑋𝑈𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
83, 5, 7syl2anc 691 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
9 fnfun 5902 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
10 fvelima 6158 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) ∈ (𝐹𝑈)) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴))
1110ex 449 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
123, 9, 113syl 18 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
13 simpl1 1057 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐽 ∈ (TopOn‘𝑋))
145sselda 3568 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑋)
15 simpl3 1059 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐴𝑋)
161kqfeq 21337 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
1713, 14, 15, 16syl3anc 1318 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
18 eleq2 2677 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
19 eleq2 2677 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
2018, 19bibi12d 334 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑤𝐴𝑤)))
2120cbvralv 3147 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤))
2217, 21syl6bb 275 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤)))
23 simpl2 1058 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑈𝐽)
24 eleq2 2677 . . . . . . . . 9 (𝑤 = 𝑈 → (𝑧𝑤𝑧𝑈))
25 eleq2 2677 . . . . . . . . 9 (𝑤 = 𝑈 → (𝐴𝑤𝐴𝑈))
2624, 25bibi12d 334 . . . . . . . 8 (𝑤 = 𝑈 → ((𝑧𝑤𝐴𝑤) ↔ (𝑧𝑈𝐴𝑈)))
2726rspcv 3278 . . . . . . 7 (𝑈𝐽 → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2823, 27syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2922, 28sylbid 229 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → (𝑧𝑈𝐴𝑈)))
30 simpr 476 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑈)
31 biimp 204 . . . . 5 ((𝑧𝑈𝐴𝑈) → (𝑧𝑈𝐴𝑈))
3229, 30, 31syl6ci 69 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3332rexlimdva 3013 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3412, 33syld 46 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → 𝐴𝑈))
358, 34impbid 201 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  wss 3540  cmpt 4643  cima 5041  Fun wfun 5798   Fn wfn 5799  cfv 5804  TopOnctopon 20518
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-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-rab 2905  df-v 3175  df-sbc 3403  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-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-fv 5812  df-topon 20523
This theorem is referenced by:  kqsat  21344  kqdisj  21345  kqcldsat  21346  kqt0lem  21349  isr0  21350  regr1lem  21352  kqreglem1  21354  kqreglem2  21355
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