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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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