Theorem cnpflfi 24345

Description: Forward direction of cnpflf 24347. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
cnpflfi	⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

Proof of Theorem cnpflfi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2763	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23592	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 487	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1	flimelbas 24314	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 ∈ ∪ 𝐽)
6	5	adantr 486	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
7	4, 6	ffvelcdmd 7106	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ∪ 𝐾)
8		simplr 781	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9		simprl 783	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		simprr 785	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (𝐹‘𝐴) ∈ 𝑥)
11		cnpimaex 23601	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
12	8, 9, 10, 11	syl3anc 1399	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
13		anass 474	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
14		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15		flimtop 24311	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
16	15	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17		toptopon2 23263	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
18	16, 17	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
19	1	flimfil 24315	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
20	19	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
21		flimopn 24321	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
22	18, 20, 21	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
23	14, 22	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿)))
24	23	simprd 501	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
25	24	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
26	25	r19.21bi 3258	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
27	26	expimpd 459	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ 𝐿))
28	27	anim1d 623	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 “ 𝑦) ⊆ 𝑥) → (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
29	13, 28	biimtrrid 246	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ((𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) → (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
30	29	reximdv2 3176	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
31	12, 30	mpd 16	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥)
32	31	expr 462	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
33	32	ralrimiva 3158	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
34		cnptop2 23588	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
35	34	adantl 487	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36		toptopon2 23263	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
37	35, 36	sylib 221	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
38		isflf 24339	. . 3 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐿 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹‘𝐴) ∈ ∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))))
39	37, 20, 4, 38	syl3anc 1399	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹‘𝐴) ∈ ∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))))
40	7, 33, 39	mpbir2and 726	1 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4870   “ cima 5658  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Topctop 23238  TopOnctopon 23255   CnP ccnp 23570  Filcfil 24191   fLim cflim 24280   fLimf cflf 24281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-map 8874  df-fbas 21702  df-fg 21703  df-top 23239  df-topon 23256  df-ntr 23365  df-nei 23443  df-cnp 23573  df-fil 24192  df-fm 24284  df-flim 24285  df-flf 24286

This theorem is used by:  cnpflf2  24346  cnpflf  24347  flfcnp  24350  cnpfcfi  24386