Theorem cnpflfi 23986

Description: Forward direction of cnpflf 23988. (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 2741	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2741	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23234	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 483	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1	flimelbas 23955	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 ∈ ∪ 𝐽)
6	5	adantr 482	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
7	4, 6	ffvelcdmd 7030	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ∪ 𝐾)
8		simplr 775	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9		simprl 777	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		simprr 779	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (𝐹‘𝐴) ∈ 𝑥)
11		cnpimaex 23243	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
12	8, 9, 10, 11	syl3anc 1380	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
13		anass 470	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
14		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15		flimtop 23952	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
16	15	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17		toptopon2 22905	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
18	16, 17	sylib 220	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
19	1	flimfil 23956	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
20	19	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
21		flimopn 23962	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
22	18, 20, 21	syl2anc 591	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿)))
24	23	simprd 497	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
25	24	adantr 482	. . . . . . . . . 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 3233	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
27	26	expimpd 455	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ 𝐿))
28	27	anim1d 618	. . . . . . 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 245	. . . . . 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 3151	. . . . 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 15	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥)
32	31	expr 458	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
33	32	ralrimiva 3133	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
34		cnptop2 23230	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
35	34	adantl 483	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36		toptopon2 22905	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
37	35, 36	sylib 220	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
38		isflf 23980	. . 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 1380	. 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 720	1 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ⊆ wss 3885  ∪ cuni 4841   “ cima 5624  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Topctop 22880  TopOnctopon 22897   CnP ccnp 23212  Filcfil 23832   fLim cflim 23921   fLimf cflf 23922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-fbas 21348  df-fg 21349  df-top 22881  df-topon 22898  df-ntr 23007  df-nei 23085  df-cnp 23215  df-fil 23833  df-fm 23925  df-flim 23926  df-flf 23927

This theorem is referenced by:  cnpflf2  23987  cnpflf  23988  flfcnp  23991  cnpfcfi  24027