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Theorem cnprcl2k 13577
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 13383 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1018 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3 simp2 998 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
4 uniexg 4438 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ V)
543ad2ant1 1018 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
6 mptexg 5740 . . . . . . 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 3818 . . . . . . . 8 (𝑗 = 𝐽 𝑗 = 𝐽)
98oveq2d 5888 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑘𝑚 𝑗) = ( 𝑘𝑚 𝐽))
10 rexeq 2673 . . . . . . . . . . 11 (𝑗 = 𝐽 → (∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦) ↔ ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)))
1110imbi2d 230 . . . . . . . . . 10 (𝑗 = 𝐽 → (((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
1211ralbidv 2477 . . . . . . . . 9 (𝑗 = 𝐽 → (∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
139, 12rabeqbidv 2732 . . . . . . . 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 4084 . . . . . . 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 3818 . . . . . . . . . 10 (𝑘 = 𝐾 𝑘 = 𝐾)
1615oveq1d 5887 . . . . . . . . 9 (𝑘 = 𝐾 → ( 𝑘𝑚 𝐽) = ( 𝐾𝑚 𝐽))
17 raleq 2672 . . . . . . . . 9 (𝑘 = 𝐾 → (∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))))
1816, 17rabeqbidv 2732 . . . . . . . 8 (𝑘 = 𝐾 → {𝑓 ∈ ( 𝑘𝑚 𝐽) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} = {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
1918mpteq2dv 4093 . . . . . . 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 13560 . . . . . . 7 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
2114, 19, 20ovmpog 6006 . . . . . 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 1238 . . . . 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 4828 . . . 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 2177 . . . . 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 5124 . . . 4 dom (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ⊆ 𝐽
2623, 25eqsstrdi 3207 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝐽)
27 toponuni 13384 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
28273ad2ant1 1018 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = 𝐽)
2926, 28sseqtrrd 3194 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝑋)
30 mptrel 4754 . . . 4 Rel (𝑥 𝐽 ↦ {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝐽 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
3122releqd 4709 . . . 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 168 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → Rel (𝐽 CnP 𝐾))
33 simp3 999 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
34 relelfvdm 5546 . . 3 ((Rel (𝐽 CnP 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
3532, 33, 34syl2anc 411 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
3629, 35sseldd 3156 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  Vcvv 2737  wss 3129   cuni 3809  cmpt 4063  dom cdm 4625  cima 4628  Rel wrel 4630  cfv 5215  (class class class)co 5872  𝑚 cmap 6645  Topctop 13366  TopOnctopon 13379   CnP ccnp 13557
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 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-topon 13380  df-cnp 13560
This theorem is referenced by:  cnpf2  13578  cnptopco  13593  cncnp  13601  cnptoprest2  13611  metcnpi  13886  metcnpi2  13887  metcnpi3  13888  limccnpcntop  14015  limccnp2lem  14016  limccnp2cntop  14017
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