MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpflfi Structured version   Visualization version   GIF version

Theorem cnpflfi 24023
Description: Forward direction of cnpflf 24025. (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 2735 . . . . 5 𝐽 = 𝐽
2 eqid 2735 . . . . 5 𝐾 = 𝐾
31, 2cnpf 23271 . . . 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 23992 . . . 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 23280 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑥𝐾 ∧ (𝐹𝐴) ∈ 𝑥) → ∃𝑦𝐽 (𝐴𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))
128, 9, 10, 11syl3anc 1370 . . . . 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 23989 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) → 𝐽 ∈ Top)
1615adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ Top)
17 toptopon2 22940 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1816, 17sylib 218 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘ 𝐽))
191flimfil 23993 . . . . . . . . . . . . . . 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 23999 . . . . . . . . . . . . . 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 3249 . . . . . . . . 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 3162 . . . . 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 3144 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝐾 ((𝐹𝐴) ∈ 𝑥 → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝑥))
34 cnptop2 23267 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
3534adantl 481 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ Top)
36 toptopon2 22940 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3735, 36sylib 218 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘ 𝐾))
38 isflf 24017 . . 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 1370 . 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 2106  wral 3059  wrex 3068  wss 3963   cuni 4912  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932   CnP ccnp 23249  Filcfil 23869   fLim cflim 23958   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-cnp 23252  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964
This theorem is referenced by:  cnpflf2  24024  cnpflf  24025  flfcnp  24028  cnpfcfi  24064
  Copyright terms: Public domain W3C validator