Theorem cnpflfi 23886

Description: Forward direction of cnpflf 23888. (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 2729	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2729	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23134	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1	flimelbas 23855	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 ∈ ∪ 𝐽)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
7	4, 6	ffvelcdmd 7057	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ∪ 𝐾)
8		simplr 768	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9		simprl 770	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		simprr 772	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (𝐹‘𝐴) ∈ 𝑥)
11		cnpimaex 23143	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
12	8, 9, 10, 11	syl3anc 1373	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
13		anass 468	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
14		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15		flimtop 23852	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
16	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17		toptopon2 22805	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
18	16, 17	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
19	1	flimfil 23856	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
21		flimopn 23862	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
22	18, 20, 21	syl2anc 584	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿)))
24	23	simprd 495	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
25	24	adantr 480	. . . . . . . . . 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 3229	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
27	26	expimpd 453	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ 𝐿))
28	27	anim1d 611	. . . . . . 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 243	. . . . . 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 3143	. . . . 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 456	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
33	32	ralrimiva 3125	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
34		cnptop2 23130	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
35	34	adantl 481	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36		toptopon2 22805	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
37	35, 36	sylib 218	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
38		isflf 23880	. . 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 1373	. 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 713	1 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∪ cuni 4871   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Topctop 22780  TopOnctopon 22797   CnP ccnp 23112  Filcfil 23732   fLim cflim 23821   fLimf cflf 23822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-fbas 21261  df-fg 21262  df-top 22781  df-topon 22798  df-ntr 22907  df-nei 22985  df-cnp 23115  df-fil 23733  df-fm 23825  df-flim 23826  df-flf 23827

This theorem is referenced by:  cnpflf2  23887  cnpflf  23888  flfcnp  23891  cnpfcfi  23927