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Theorem cnprcl2k 13709
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
Assertion
Ref Expression
cnprcl2k ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ 𝑋)
Proof of Theorem cnprcl2k
Dummy variables π‘₯ 𝑓 𝑔 𝑗 π‘˜ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 13517 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
213ad2ant1 1018 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐽 ∈ Top)
3 simp2 998 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐾 ∈ Top)
4 uniexg 4440 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 ∈ V)
543ad2ant1 1018 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ βˆͺ 𝐽 ∈ V)
6 mptexg 5742 . . . . . . 7 (βˆͺ 𝐽 ∈ V β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) ∈ V)
75, 6syl 14 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) ∈ V)
8 unieq 3819 . . . . . . . 8 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
98oveq2d 5891 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) = (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽))
10 rexeq 2674 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ (βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦) ↔ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦)))
1110imbi2d 230 . . . . . . . . . 10 (𝑗 = 𝐽 β†’ (((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦)) ↔ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))))
1211ralbidv 2477 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦)) ↔ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))))
139, 12rabeqbidv 2733 . . . . . . . 8 (𝑗 = 𝐽 β†’ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))} = {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))})
148, 13mpteq12dv 4086 . . . . . . 7 (𝑗 = 𝐽 β†’ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
15 unieq 3819 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
1615oveq1d 5890 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽) = (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽))
17 raleq 2673 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦)) ↔ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))))
1816, 17rabeqbidv 2733 . . . . . . . 8 (π‘˜ = 𝐾 β†’ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))} = {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))})
1918mpteq2dv 4095 . . . . . . 7 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
20 df-cnp 13692 . . . . . . 7 CnP = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝑗 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
2114, 19, 20ovmpog 6009 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) ∈ V) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
222, 3, 7, 21syl3anc 1238 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
2322dmeqd 4830 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ dom (𝐽 CnP 𝐾) = dom (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}))
24 eqid 2177 . . . . 5 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))})
2524dmmptss 5126 . . . 4 dom (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))}) βŠ† βˆͺ 𝐽
2623, 25eqsstrdi 3208 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ dom (𝐽 CnP 𝐾) βŠ† βˆͺ 𝐽)
27 toponuni 13518 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
28273ad2ant1 1018 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑋 = βˆͺ 𝐽)
2926, 28sseqtrrd 3195 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ dom (𝐽 CnP 𝐾) βŠ† 𝑋)
30 mptrel 4756 . . . 4 Rel (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))})
3122releqd 4711 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ (Rel (𝐽 CnP 𝐾) ↔ Rel (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑓 ∈ (βˆͺ 𝐾 β†‘π‘š βˆͺ 𝐽) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘” ∈ 𝐽 (π‘₯ ∈ 𝑔 ∧ (𝑓 β€œ 𝑔) βŠ† 𝑦))})))
3230, 31mpbiri 168 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ Rel (𝐽 CnP 𝐾))
33 simp3 999 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))
34 relelfvdm 5548 . . 3 ((Rel (𝐽 CnP 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ dom (𝐽 CnP 𝐾))
3532, 33, 34syl2anc 411 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ dom (𝐽 CnP 𝐾))
3629, 35sseldd 3157 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ 𝑋)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  {crab 2459  Vcvv 2738   βŠ† wss 3130  βˆͺ cuni 3810   ↦ cmpt 4065  dom cdm 4627   β€œ cima 4630  Rel wrel 4632  β€˜cfv 5217  (class class class)co 5875   β†‘π‘š cmap 6648  Topctop 13500  TopOnctopon 13513   CnP ccnp 13689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-topon 13514  df-cnp 13692
This theorem is referenced by:  cnpf2  13710  cnptopco  13725  cncnp  13733  cnptoprest2  13743  metcnpi  14018  metcnpi2  14019  metcnpi3  14020  limccnpcntop  14147  limccnp2lem  14148  limccnp2cntop  14149
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