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Theorem cnprcl2k 12553
 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 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)

Proof of Theorem cnprcl2k
Dummy variables 𝑥 𝑓 𝑔 𝑗 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 12359 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1003 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3 simp2 983 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
4 uniexg 4394 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ V)
543ad2ant1 1003 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
6 mptexg 5685 . . . . . . 7 ( 𝐽 ∈ V → (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V)
75, 6syl 14 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V)
8 unieq 3777 . . . . . . . 8 (𝑗 = 𝐽 𝑗 = 𝐽)
98oveq2d 5830 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑘𝑚 𝑗) = ( 𝑘𝑚 𝐽))
10 rexeq 2650 . . . . . . . . . . 11 (𝑗 = 𝐽 → (∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦) ↔ ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)))
1110imbi2d 229 . . . . . . . . . 10 (𝑗 = 𝐽 → (((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
1211ralbidv 2454 . . . . . . . . 9 (𝑗 = 𝐽 → (∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
139, 12rabeqbidv 2704 . . . . . . . 8 (𝑗 = 𝐽 → {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} = {𝑓 ∈ ( 𝑘𝑚 𝐽) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
148, 13mpteq12dv 4042 . . . . . . 7 (𝑗 = 𝐽 → (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) = (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝑘𝑚 𝐽) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
15 unieq 3777 . . . . . . . . . 10 (𝑘 = 𝐾 𝑘 = 𝐾)
1615oveq1d 5829 . . . . . . . . 9 (𝑘 = 𝐾 → ( 𝑘𝑚 𝐽) = ( 𝐾𝑚 𝐽))
17 raleq 2649 . . . . . . . . 9 (𝑘 = 𝐾 → (∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
1816, 17rabeqbidv 2704 . . . . . . . 8 (𝑘 = 𝐾 → {𝑓 ∈ ( 𝑘𝑚 𝐽) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} = {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
1918mpteq2dv 4051 . . . . . . 7 (𝑘 = 𝐾 → (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝑘𝑚 𝐽) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) = (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
20 df-cnp 12536 . . . . . . 7 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
2114, 19, 20ovmpog 5945 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V) → (𝐽 CnP 𝐾) = (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
222, 3, 7, 21syl3anc 1217 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽 CnP 𝐾) = (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
2322dmeqd 4781 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) = dom (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
24 eqid 2154 . . . . 5 (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) = (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
2524dmmptss 5075 . . . 4 dom (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ⊆ 𝐽
2623, 25eqsstrdi 3176 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝐽)
27 toponuni 12360 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
28273ad2ant1 1003 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = 𝐽)
2926, 28sseqtrrd 3163 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝑋)
30 mptrel 4707 . . . 4 Rel (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
3122releqd 4663 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (Rel (𝐽 CnP 𝐾) ↔ Rel (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})))
3230, 31mpbiri 167 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → Rel (𝐽 CnP 𝐾))
33 simp3 984 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
34 relelfvdm 5493 . . 3 ((Rel (𝐽 CnP 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
3532, 33, 34syl2anc 409 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
3629, 35sseldd 3125 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  {crab 2436  Vcvv 2709   ⊆ wss 3098  ∪ cuni 3768   ↦ cmpt 4021  dom cdm 4579   “ cima 4582  Rel wrel 4584  ‘cfv 5163  (class class class)co 5814   ↑𝑚 cmap 6582  Topctop 12342  TopOnctopon 12355   CnP ccnp 12533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-topon 12356  df-cnp 12536 This theorem is referenced by:  cnpf2  12554  cnptopco  12569  cncnp  12577  cnptoprest2  12587  metcnpi  12862  metcnpi2  12863  metcnpi3  12864  limccnpcntop  12991  limccnp2lem  12992  limccnp2cntop  12993
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