Theorem cnprcl2k 15000

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)) → 𝑃 ∈ 𝑋)

Proof of Theorem cnprcl2k

Dummy variables 𝑥 𝑓 𝑔 𝑗 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 14808	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1045	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3		simp2 1025	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
4		uniexg 4542	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ∈ V)
5	4	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∪ 𝐽 ∈ V)
6		mptexg 5889	. . . . . . 7 ⊢ (∪ 𝐽 ∈ V → (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V)
7	5, 6	syl 14	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V)
8		unieq 3907	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
9	8	oveq2d 6044	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑𝑚 ∪ 𝑗) = (∪ 𝑘 ↑𝑚 ∪ 𝐽))
10		rexeq 2732	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦) ↔ ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦)))
11	10	imbi2d 230	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦)) ↔ ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))))
12	11	ralbidv 2533	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))))
13	9, 12	rabeqbidv 2798	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} = {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
14	8, 13	mpteq12dv 4176	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
15		unieq 3907	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
16	15	oveq1d 6043	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑𝑚 ∪ 𝐽) = (∪ 𝐾 ↑𝑚 ∪ 𝐽))
17		raleq 2731	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))))
18	16, 17	rabeqbidv 2798	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
19	18	mpteq2dv 4185	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
20		df-cnp 14983	. . . . . . 7 ⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
21	14, 19, 20	ovmpog 6166	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V) → (𝐽 CnP 𝐾) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
22	2, 3, 7, 21	syl3anc 1274	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽 CnP 𝐾) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
23	22	dmeqd 4939	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) = dom (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
24		eqid 2231	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
25	24	dmmptss 5240	. . . 4 ⊢ dom (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ⊆ ∪ 𝐽
26	23, 25	eqsstrdi 3280	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ ∪ 𝐽)
27		toponuni 14809	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	27	3ad2ant1 1045	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = ∪ 𝐽)
29	26, 28	sseqtrrd 3267	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝑋)
30		mptrel 4864	. . . 4 ⊢ Rel (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
31	22	releqd 4816	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (Rel (𝐽 CnP 𝐾) ↔ Rel (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})))
32	30, 31	mpbiri 168	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → Rel (𝐽 CnP 𝐾))
33		simp3 1026	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
34		relelfvdm 5680	. . 3 ⊢ ((Rel (𝐽 CnP 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
35	32, 33, 34	syl2anc 411	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
36	29, 35	sseldd 3229	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  Vcvv 2803   ⊆ wss 3201  ∪ cuni 3898   ↦ cmpt 4155  dom cdm 4731   “ cima 4734  Rel wrel 4736  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860  Topctop 14791  TopOnctopon 14804   CnP ccnp 14980

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-topon 14805  df-cnp 14983

This theorem is referenced by:  cnpf2  15001  cnptopco  15016  cncnp  15024  cnptoprest2  15034  metcnpi  15309  metcnpi2  15310  metcnpi3  15311  limccnpcntop  15469  limccnp2lem  15470  limccnp2cntop  15471