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Theorem cnpflfi 22607
 Description: Forward direction of cnpflf 22609. (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)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

Proof of Theorem cnpflfi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . . 5 𝐽 = 𝐽
2 eqid 2801 . . . . 5 𝐾 = 𝐾
31, 2cnpf 21855 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
43adantl 485 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
51flimelbas 22576 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 𝐽)
65adantr 484 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
74, 6ffvelrnd 6833 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ 𝐾)
8 simplr 768 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9 simprl 770 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝑥𝐾)
10 simprr 772 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (𝐹𝐴) ∈ 𝑥)
11 cnpimaex 21864 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
128, 9, 10, 11syl3anc 1368 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
13 anass 472 . . . . . . 7 (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) ↔ (𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))
14 simpl 486 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15 flimtop 22573 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
1615adantr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17 toptopon2 21526 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1816, 17sylib 221 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘ 𝐽))
191flimfil 22577 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
2019adantr 484 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
21 flimopn 22583 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2218, 20, 21syl2anc 587 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2314, 22mpbid 235 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿)))
2423simprd 499 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2524adantr 484 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2625r19.21bi 3176 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) ∧ 𝑦𝐽) → (𝐴𝑦𝑦𝐿))
2726expimpd 457 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽𝐴𝑦) → 𝑦𝐿))
2827anim1d 613 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
2913, 28syl5bir 246 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
3029reximdv2 3233 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
3112, 30mpd 15 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥)
3231expr 460 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝐾) → ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
3332ralrimiva 3152 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
34 cnptop2 21851 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
3534adantl 485 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36 toptopon2 21526 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3735, 36sylib 221 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘ 𝐾))
38 isflf 22601 . . 3 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝐾 ∧ ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))))
3937, 20, 4, 38syl3anc 1368 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝐾 ∧ ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))))
407, 33, 39mpbir2and 712 1 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110   ⊆ wss 3884  ∪ cuni 4803   “ cima 5526  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Topctop 21501  TopOnctopon 21518   CnP ccnp 21833  Filcfil 22453   fLim cflim 22542   fLimf cflf 22543 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-fbas 20091  df-fg 20092  df-top 21502  df-topon 21519  df-ntr 21628  df-nei 21706  df-cnp 21836  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548 This theorem is referenced by:  cnpflf2  22608  cnpflf  22609  flfcnp  22612  cnpfcfi  22648
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