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Theorem kqfvima 21754
 Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 6639, 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 21749 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1127 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝐹 Fn 𝑋)
4 toponss 20952 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝑋)
543adant3 1126 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → 𝑈𝑋)
6 fnfvima 6639 . . . 4 ((𝐹 Fn 𝑋𝑈𝑋𝐴𝑈) → (𝐹𝐴) ∈ (𝐹𝑈))
763expia 1114 . . 3 ((𝐹 Fn 𝑋𝑈𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
83, 5, 7syl2anc 573 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 → (𝐹𝐴) ∈ (𝐹𝑈)))
9 fnfun 6128 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
10 fvelima 6390 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) ∈ (𝐹𝑈)) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴))
1110ex 397 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
123, 9, 113syl 18 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → ∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴)))
13 simpl1 1227 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐽 ∈ (TopOn‘𝑋))
145sselda 3752 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑋)
15 simpl3 1231 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝐴𝑋)
161kqfeq 21748 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
1713, 14, 15, 16syl3anc 1476 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
18 eleq2 2839 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
19 eleq2 2839 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
2018, 19bibi12d 334 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑤𝐴𝑤)))
2120cbvralv 3320 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤))
2217, 21syl6bb 276 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑤𝐽 (𝑧𝑤𝐴𝑤)))
23 simpl2 1229 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑈𝐽)
24 eleq2 2839 . . . . . . . . 9 (𝑤 = 𝑈 → (𝑧𝑤𝑧𝑈))
25 eleq2 2839 . . . . . . . . 9 (𝑤 = 𝑈 → (𝐴𝑤𝐴𝑈))
2624, 25bibi12d 334 . . . . . . . 8 (𝑤 = 𝑈 → ((𝑧𝑤𝐴𝑤) ↔ (𝑧𝑈𝐴𝑈)))
2726rspcv 3456 . . . . . . 7 (𝑈𝐽 → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2823, 27syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → (∀𝑤𝐽 (𝑧𝑤𝐴𝑤) → (𝑧𝑈𝐴𝑈)))
2922, 28sylbid 230 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → (𝑧𝑈𝐴𝑈)))
30 simpr 471 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → 𝑧𝑈)
31 biimp 205 . . . . 5 ((𝑧𝑈𝐴𝑈) → (𝑧𝑈𝐴𝑈))
3229, 30, 31syl6ci 71 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) ∧ 𝑧𝑈) → ((𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3332rexlimdva 3179 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (∃𝑧𝑈 (𝐹𝑧) = (𝐹𝐴) → 𝐴𝑈))
3412, 33syld 47 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → ((𝐹𝐴) ∈ (𝐹𝑈) → 𝐴𝑈))
358, 34impbid 202 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065   ⊆ wss 3723   ↦ cmpt 4863   “ cima 5252  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031  TopOnctopon 20935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-topon 20936 This theorem is referenced by:  kqsat  21755  kqdisj  21756  kqcldsat  21757  kqt0lem  21760  isr0  21761  regr1lem  21763  kqreglem1  21765  kqreglem2  21766
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