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Theorem kqfvima 23084
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 7184, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqfvima ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ π‘ˆ ↔ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ)))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐽,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)

Proof of Theorem kqfvima
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqffn 23079 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
323ad2ant1 1134 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 Fn 𝑋)
4 toponss 22279 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽) β†’ π‘ˆ βŠ† 𝑋)
543adant3 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ π‘ˆ βŠ† 𝑋)
6 fnfvima 7184 . . . 4 ((𝐹 Fn 𝑋 ∧ π‘ˆ βŠ† 𝑋 ∧ 𝐴 ∈ π‘ˆ) β†’ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ))
763expia 1122 . . 3 ((𝐹 Fn 𝑋 ∧ π‘ˆ βŠ† 𝑋) β†’ (𝐴 ∈ π‘ˆ β†’ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ)))
83, 5, 7syl2anc 585 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ π‘ˆ β†’ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ)))
9 fnfun 6603 . . . 4 (𝐹 Fn 𝑋 β†’ Fun 𝐹)
10 fvelima 6909 . . . . 5 ((Fun 𝐹 ∧ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ)) β†’ βˆƒπ‘§ ∈ π‘ˆ (πΉβ€˜π‘§) = (πΉβ€˜π΄))
1110ex 414 . . . 4 (Fun 𝐹 β†’ ((πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ) β†’ βˆƒπ‘§ ∈ π‘ˆ (πΉβ€˜π‘§) = (πΉβ€˜π΄)))
123, 9, 113syl 18 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ ((πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ) β†’ βˆƒπ‘§ ∈ π‘ˆ (πΉβ€˜π‘§) = (πΉβ€˜π΄)))
13 simpl1 1192 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
145sselda 3945 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ 𝑋)
15 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ 𝐴 ∈ 𝑋)
161kqfeq 23078 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
1713, 14, 15, 16syl3anc 1372 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
18 eleq2 2827 . . . . . . . . 9 (𝑦 = 𝑀 β†’ (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑀))
19 eleq2 2827 . . . . . . . . 9 (𝑦 = 𝑀 β†’ (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑀))
2018, 19bibi12d 346 . . . . . . . 8 (𝑦 = 𝑀 β†’ ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀)))
2120cbvralvw 3226 . . . . . . 7 (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ βˆ€π‘€ ∈ 𝐽 (𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
2217, 21bitrdi 287 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘€ ∈ 𝐽 (𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀)))
23 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ π‘ˆ ∈ 𝐽)
24 eleq2 2827 . . . . . . . . 9 (𝑀 = π‘ˆ β†’ (𝑧 ∈ 𝑀 ↔ 𝑧 ∈ π‘ˆ))
25 eleq2 2827 . . . . . . . . 9 (𝑀 = π‘ˆ β†’ (𝐴 ∈ 𝑀 ↔ 𝐴 ∈ π‘ˆ))
2624, 25bibi12d 346 . . . . . . . 8 (𝑀 = π‘ˆ β†’ ((𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀) ↔ (𝑧 ∈ π‘ˆ ↔ 𝐴 ∈ π‘ˆ)))
2726rspcv 3578 . . . . . . 7 (π‘ˆ ∈ 𝐽 β†’ (βˆ€π‘€ ∈ 𝐽 (𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀) β†’ (𝑧 ∈ π‘ˆ ↔ 𝐴 ∈ π‘ˆ)))
2823, 27syl 17 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ (βˆ€π‘€ ∈ 𝐽 (𝑧 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀) β†’ (𝑧 ∈ π‘ˆ ↔ 𝐴 ∈ π‘ˆ)))
2922, 28sylbid 239 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) β†’ (𝑧 ∈ π‘ˆ ↔ 𝐴 ∈ π‘ˆ)))
30 simpr 486 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ π‘ˆ)
31 biimp 214 . . . . 5 ((𝑧 ∈ π‘ˆ ↔ 𝐴 ∈ π‘ˆ) β†’ (𝑧 ∈ π‘ˆ β†’ 𝐴 ∈ π‘ˆ))
3229, 30, 31syl6ci 71 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ π‘ˆ) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) β†’ 𝐴 ∈ π‘ˆ))
3332rexlimdva 3153 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘§ ∈ π‘ˆ (πΉβ€˜π‘§) = (πΉβ€˜π΄) β†’ 𝐴 ∈ π‘ˆ))
3412, 33syld 47 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ ((πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ) β†’ 𝐴 ∈ π‘ˆ))
358, 34impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ π‘ˆ ↔ (πΉβ€˜π΄) ∈ (𝐹 β€œ π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3408   βŠ† wss 3911   ↦ cmpt 5189   β€œ cima 5637  Fun wfun 6491   Fn wfn 6492  β€˜cfv 6497  TopOnctopon 22262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-topon 22263
This theorem is referenced by:  kqsat  23085  kqdisj  23086  kqcldsat  23087  kqt0lem  23090  isr0  23091  regr1lem  23093  kqreglem1  23095  kqreglem2  23096
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