Theorem cnprcl2k 14929

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 14737	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1044	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3		simp2 1024	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
4		uniexg 4536	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ∈ V)
5	4	3ad2ant1 1044	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∪ 𝐽 ∈ V)
6		mptexg 5878	. . . . . . 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 3902	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
9	8	oveq2d 6033	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑𝑚 ∪ 𝑗) = (∪ 𝑘 ↑𝑚 ∪ 𝐽))
10		rexeq 2731	. . . . . . . . . . 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 2532	. . . . . . . . 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 2797	. . . . . . . 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 4171	. . . . . . 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 3902	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
16	15	oveq1d 6032	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑𝑚 ∪ 𝐽) = (∪ 𝐾 ↑𝑚 ∪ 𝐽))
17		raleq 2730	. . . . . . . . 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 2797	. . . . . . . 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 4180	. . . . . . 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 14912	. . . . . . 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 6155	. . . . . 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 1273	. . . . 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 4933	. . . 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 5233	. . . 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 3279	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ ∪ 𝐽)
27		toponuni 14738	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	27	3ad2ant1 1044	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = ∪ 𝐽)
29	26, 28	sseqtrrd 3266	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝑋)
30		mptrel 4858	. . . 4 ⊢ Rel (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
31	22	releqd 4810	. . . 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 1025	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
34		relelfvdm 5671	. . 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 3228	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514  Vcvv 2802   ⊆ wss 3200  ∪ cuni 3893   ↦ cmpt 4150  dom cdm 4725   “ cima 4728  Rel wrel 4730  ‘cfv 5326  (class class class)co 6017   ↑_𝑚 cmap 6816  Topctop 14720  TopOnctopon 14733   CnP ccnp 14909

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-topon 14734  df-cnp 14912

This theorem is referenced by:  cnpf2  14930  cnptopco  14945  cncnp  14953  cnptoprest2  14963  metcnpi  15238  metcnpi2  15239  metcnpi3  15240  limccnpcntop  15398  limccnp2lem  15399  limccnp2cntop  15400