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Theorem cnpflfi 23502
Description: Forward direction of cnpflf 23504. (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)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))

Proof of Theorem cnpflfi
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2732 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnpf 22750 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
43adantl 482 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
51flimelbas 23471 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐴 ∈ βˆͺ 𝐽)
65adantr 481 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ βˆͺ 𝐽)
74, 6ffvelcdmd 7087 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ βˆͺ 𝐾)
8 simplr 767 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
9 simprl 769 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ π‘₯ ∈ 𝐾)
10 simprr 771 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ (πΉβ€˜π΄) ∈ π‘₯)
11 cnpimaex 22759 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ∧ π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯))
128, 9, 10, 11syl3anc 1371 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯))
13 anass 469 . . . . . . 7 (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯)))
14 simpl 483 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ (𝐽 fLim 𝐿))
15 flimtop 23468 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐽 ∈ Top)
1615adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐽 ∈ Top)
17 toptopon2 22419 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1816, 17sylib 217 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
191flimfil 23472 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
2019adantr 481 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
21 flimopn 23478 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))))
2218, 20, 21syl2anc 584 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐴 ∈ (𝐽 fLim 𝐿) ↔ (𝐴 ∈ βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))))
2314, 22mpbid 231 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐴 ∈ βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿)))
2423simprd 496 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))
2524adantr 481 . . . . . . . . . 10 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))
2625r19.21bi 3248 . . . . . . . . 9 ((((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) ∧ 𝑦 ∈ 𝐽) β†’ (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))
2726expimpd 454 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ ((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) β†’ 𝑦 ∈ 𝐿))
2827anim1d 611 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯) β†’ (𝑦 ∈ 𝐿 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯)))
2913, 28biimtrrid 242 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ ((𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯)) β†’ (𝑦 ∈ 𝐿 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯)))
3029reximdv2 3164 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ (βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯))
3112, 30mpd 15 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯)
3231expr 457 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ π‘₯ ∈ 𝐾) β†’ ((πΉβ€˜π΄) ∈ π‘₯ β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯))
3332ralrimiva 3146 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘₯ ∈ 𝐾 ((πΉβ€˜π΄) ∈ π‘₯ β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯))
34 cnptop2 22746 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐾 ∈ Top)
3534adantl 482 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐾 ∈ Top)
36 toptopon2 22419 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
3735, 36sylib 217 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
38 isflf 23496 . . 3 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ) ↔ ((πΉβ€˜π΄) ∈ βˆͺ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 ((πΉβ€˜π΄) ∈ π‘₯ β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯))))
3937, 20, 4, 38syl3anc 1371 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ) ↔ ((πΉβ€˜π΄) ∈ βˆͺ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 ((πΉβ€˜π΄) ∈ π‘₯ β†’ βˆƒπ‘¦ ∈ 𝐿 (𝐹 β€œ 𝑦) βŠ† π‘₯))))
407, 33, 39mpbir2and 711 1 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  βˆͺ cuni 4908   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Topctop 22394  TopOnctopon 22411   CnP ccnp 22728  Filcfil 23348   fLim cflim 23437   fLimf cflf 23438
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-fbas 20940  df-fg 20941  df-top 22395  df-topon 22412  df-ntr 22523  df-nei 22601  df-cnp 22731  df-fil 23349  df-fm 23441  df-flim 23442  df-flf 23443
This theorem is referenced by:  cnpflf2  23503  cnpflf  23504  flfcnp  23507  cnpfcfi  23543
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