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Theorem cnpflfi 24007
Description: Forward direction of cnpflf 24009. (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 2737 . . . . 5 𝐽 = 𝐽
2 eqid 2737 . . . . 5 𝐾 = 𝐾
31, 2cnpf 23255 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
43adantl 481 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
51flimelbas 23976 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 𝐽)
65adantr 480 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
74, 6ffvelcdmd 7105 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ 𝐾)
8 simplr 769 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9 simprl 771 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝑥𝐾)
10 simprr 773 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (𝐹𝐴) ∈ 𝑥)
11 cnpimaex 23264 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
128, 9, 10, 11syl3anc 1373 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
13 anass 468 . . . . . . 7 (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) ↔ (𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))
14 simpl 482 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15 flimtop 23973 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
1615adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17 toptopon2 22924 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1816, 17sylib 218 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘ 𝐽))
191flimfil 23977 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
2019adantr 480 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
21 flimopn 23983 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2218, 20, 21syl2anc 584 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2314, 22mpbid 232 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿)))
2423simprd 495 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2524adantr 480 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2625r19.21bi 3251 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) ∧ 𝑦𝐽) → (𝐴𝑦𝑦𝐿))
2726expimpd 453 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽𝐴𝑦) → 𝑦𝐿))
2827anim1d 611 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
2913, 28biimtrrid 243 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
3029reximdv2 3164 . . . . 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 456 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝐾) → ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
3332ralrimiva 3146 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
34 cnptop2 23251 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
3534adantl 481 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36 toptopon2 22924 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3735, 36sylib 218 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘ 𝐾))
38 isflf 24001 . . 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 1373 . 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 713 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 206  wa 395  wcel 2108  wral 3061  wrex 3070  wss 3951   cuni 4907  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916   CnP ccnp 23233  Filcfil 23853   fLim cflim 23942   fLimf cflf 23943
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-ntr 23028  df-nei 23106  df-cnp 23236  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948
This theorem is referenced by:  cnpflf2  24008  cnpflf  24009  flfcnp  24012  cnpfcfi  24048
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