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Theorem cnpflfi 22314
Description: Forward direction of cnpflf 22316. (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 2778 . . . . 5 𝐽 = 𝐽
2 eqid 2778 . . . . 5 𝐾 = 𝐾
31, 2cnpf 21562 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
43adantl 474 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
51flimelbas 22283 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐴 𝐽)
65adantr 473 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
74, 6ffvelrnd 6679 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ 𝐾)
8 simplr 756 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
9 simprl 758 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → 𝑥𝐾)
10 simprr 760 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (𝐹𝐴) ∈ 𝑥)
11 cnpimaex 21571 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
128, 9, 10, 11syl3anc 1351 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
13 anass 461 . . . . . . 7 (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) ↔ (𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))
14 simpl 475 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
15 flimtop 22280 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
1615adantr 473 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17 toptopon2 21233 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1816, 17sylib 210 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘ 𝐽))
191flimfil 22284 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
2019adantr 473 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
21 flimopn 22290 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2218, 20, 21syl2anc 576 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))))
2314, 22mpbid 224 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 𝐽 ∧ ∀𝑦𝐽 (𝐴𝑦𝑦𝐿)))
2423simprd 488 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2524adantr 473 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ∀𝑦𝐽 (𝐴𝑦𝑦𝐿))
2625r19.21bi 3158 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) ∧ 𝑦𝐽) → (𝐴𝑦𝑦𝐿))
2726expimpd 446 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽𝐴𝑦) → 𝑦𝐿))
2827anim1d 601 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → (((𝑦𝐽𝐴𝑦) ∧ (𝐹𝑦) ⊆ 𝑥) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
2913, 28syl5bir 235 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥)) → ((𝑦𝐽 ∧ (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) → (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝑥)))
3029reximdv2 3216 . . . . 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 449 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝐾) → ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
3332ralrimiva 3132 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
34 cnptop2 21558 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
3534adantl 474 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36 toptopon2 21233 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3735, 36sylib 210 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘ 𝐾))
38 isflf 22308 . . 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 1351 . 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 700 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 198  wa 387  wcel 2050  wral 3088  wrex 3089  wss 3831   cuni 4713  cima 5411  wf 6186  cfv 6190  (class class class)co 6978  Topctop 21208  TopOnctopon 21225   CnP ccnp 21540  Filcfil 22160   fLim cflim 22249   fLimf cflf 22250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-map 8210  df-fbas 20247  df-fg 20248  df-top 21209  df-topon 21226  df-ntr 21335  df-nei 21413  df-cnp 21543  df-fil 22161  df-fm 22253  df-flim 22254  df-flf 22255
This theorem is referenced by:  cnpflf2  22315  cnpflf  22316  flfcnp  22319  cnpfcfi  22355
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