Theorem cnprcl2k 14526

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 14334	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1020	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3		simp2 1000	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
4		uniexg 4475	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ∈ V)
5	4	3ad2ant1 1020	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∪ 𝐽 ∈ V)
6		mptexg 5790	. . . . . . 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 3849	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
9	8	oveq2d 5941	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑𝑚 ∪ 𝑗) = (∪ 𝑘 ↑𝑚 ∪ 𝐽))
10		rexeq 2694	. . . . . . . . . . 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 2497	. . . . . . . . 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 2758	. . . . . . . 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 4116	. . . . . . 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 3849	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
16	15	oveq1d 5940	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑𝑚 ∪ 𝐽) = (∪ 𝐾 ↑𝑚 ∪ 𝐽))
17		raleq 2693	. . . . . . . . 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 2758	. . . . . . . 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 4125	. . . . . . 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 14509	. . . . . . 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 6061	. . . . . 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 1249	. . . . 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 4869	. . . 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 2196	. . . . 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 5167	. . . 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 3236	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ ∪ 𝐽)
27		toponuni 14335	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	27	3ad2ant1 1020	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = ∪ 𝐽)
29	26, 28	sseqtrrd 3223	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → dom (𝐽 CnP 𝐾) ⊆ 𝑋)
30		mptrel 4795	. . . 4 ⊢ Rel (𝑥 ∈ ∪ 𝐽 ↦ {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝐽 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
31	22	releqd 4748	. . . 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 1001	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
34		relelfvdm 5593	. . 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 3185	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3840   ↦ cmpt 4095  dom cdm 4664   “ cima 4667  Rel wrel 4669  ‘cfv 5259  (class class class)co 5925   ↑_𝑚 cmap 6716  Topctop 14317  TopOnctopon 14330   CnP ccnp 14506

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-topon 14331  df-cnp 14509

This theorem is referenced by:  cnpf2  14527  cnptopco  14542  cncnp  14550  cnptoprest2  14560  metcnpi  14835  metcnpi2  14836  metcnpi3  14837  limccnpcntop  14995  limccnp2lem  14996  limccnp2cntop  14997