Theorem cnpflfi 23964

Description: Forward direction of cnpflf 23966. (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 2736	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2736	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23212	. . . 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 23933	. . . 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 7037	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ∪ 𝐾)
8		simplr 769	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9		simprl 771	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		simprr 773	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (𝐹‘𝐴) ∈ 𝑥)
11		cnpimaex 23221	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
12	8, 9, 10, 11	syl3anc 1374	. . . . 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 23930	. . . . . . . . . . . . . . . 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 22883	. . . . . . . . . . . . . . 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 23934	. . . . . . . . . . . . . . 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 23940	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
22	18, 20, 21	syl2anc 585	. . . . . . . . . . . . 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 612	. . . . . . 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 3147	. . . . 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 3129	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
34		cnptop2 23208	. . . . 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 22883	. . . 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 23958	. . 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 1374	. 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 714	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 2114  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  ∪ cuni 4850   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Topctop 22858  TopOnctopon 22875   CnP ccnp 23190  Filcfil 23810   fLim cflim 23899   fLimf cflf 23900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-ntr 22985  df-nei 23063  df-cnp 23193  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905

This theorem is referenced by:  cnpflf2  23965  cnpflf  23966  flfcnp  23969  cnpfcfi  24005