Theorem cnpflfi 24165

Description: Forward direction of cnpflf 24167. (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 23413	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 486	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1	flimelbas 24134	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 ∈ ∪ 𝐽)
6	5	adantr 485	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
7	4, 6	ffvelcdmd 7080	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ∪ 𝐾)
8		simplr 780	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9		simprl 782	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		simprr 784	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → (𝐹‘𝐴) ∈ 𝑥)
11		cnpimaex 23422	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
12	8, 9, 10, 11	syl3anc 1398	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))
13		anass 473	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))
14		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15		flimtop 24131	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
16	15	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17		toptopon2 23084	. . . . . . . . . . . . . . 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 24135	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
20	19	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
21		flimopn 24141	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))))
22	18, 20, 21	syl2anc 595	. . . . . . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
25	24	adantr 485	. . . . . . . . . 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 3257	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → 𝑦 ∈ 𝐿))
27	26	expimpd 458	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑥)) → ((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ 𝐿))
28	27	anim1d 622	. . . . . . 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 3175	. . . . 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 461	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
33	32	ralrimiva 3157	. 2 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝑥))
34		cnptop2 23409	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
35	34	adantl 486	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36		toptopon2 23084	. . . 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 24159	. . 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 1398	. 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 725	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 400   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ∪ cuni 4872   “ cima 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Topctop 23059  TopOnctopon 23076   CnP ccnp 23391  Filcfil 24011   fLim cflim 24100   fLimf cflf 24101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-fbas 21528  df-fg 21529  df-top 23060  df-topon 23077  df-ntr 23186  df-nei 23264  df-cnp 23394  df-fil 24012  df-fm 24104  df-flim 24105  df-flf 24106

This theorem is used by:  cnpflf2  24166  cnpflf  24167  flfcnp  24170  cnpfcfi  24206