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Theorem kqfvima 23754
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 23749 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1132 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝐹 Fn 𝑋)
4 toponss 22949 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
543adant3 1131 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝑈𝑋)
6 fnfvima 7253 . . . 4 ((𝐹 Fn 𝑋𝑈𝑋𝐴𝑈) → (𝐹𝐴) ∈ (𝐹𝑈))
763expia 1120 . . 3 ((𝐹 Fn 𝑋𝑈𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
83, 5, 7syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
9 fnfun 6669 . . . 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 1190 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐽 ∈ (TopOn‘𝑋))
145sselda 3995 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑋)
15 simpl3 1192 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐴𝑋)
161kqfeq 23748 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
1713, 14, 15, 16syl3anc 1370 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
18 eleq2 2828 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
19 eleq2 2828 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
2018, 19bibi12d 345 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑤𝐴𝑤)))
2120cbvralvw 3235 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤))
2217, 21bitrdi 287 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤)))
23 simpl2 1191 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑈𝐽)
24 eleq2 2828 . . . . . . . . 9 (𝑤 = 𝑈 → (𝑧𝑤𝑧𝑈))
25 eleq2 2828 . . . . . . . . 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 3153 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963  cmpt 5231  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-topon 22933
This theorem is referenced by:  kqsat  23755  kqdisj  23756  kqcldsat  23757  kqt0lem  23760  isr0  23761  regr1lem  23763  kqreglem1  23765  kqreglem2  23766
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