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Theorem cnpflfi 23373
Description: Forward direction of cnpflf 23375. (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 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2733 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
31, 2cnpf 22621 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
43adantl 483 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
51flimelbas 23342 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐴 ∈ βˆͺ 𝐽)
65adantr 482 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ βˆͺ 𝐽)
74, 6ffvelcdmd 7040 . 2 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ βˆͺ 𝐾)
8 simplr 768 . . . . . 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 770 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ π‘₯ ∈ 𝐾)
10 simprr 772 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π΄) ∈ π‘₯)) β†’ (πΉβ€˜π΄) ∈ π‘₯)
11 cnpimaex 22630 . . . . . 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 1372 . . . . 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 470 . . . . . . 7 (((𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯) ↔ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ (𝐹 β€œ 𝑦) βŠ† π‘₯)))
14 simpl 484 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ (𝐽 fLim 𝐿))
15 flimtop 23339 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐽 ∈ Top)
1615adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐽 ∈ Top)
17 toptopon2 22290 . . . . . . . . . . . . . . 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 23343 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝐽 fLim 𝐿) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
2019adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
21 flimopn 23349 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . 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 497 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ 𝑦 ∈ 𝐿))
2524adantr 482 . . . . . . . . . 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 3233 . . . . . . . . 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 455 . . . . . . . 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 612 . . . . . . 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 3158 . . . . 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 458 . . 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 3140 . 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 22617 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐾 ∈ Top)
3534adantl 483 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐾 ∈ Top)
36 toptopon2 22290 . . . 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 23367 . . 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 1372 . 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 712 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 397   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914  βˆͺ cuni 4869   β€œ cima 5640  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Topctop 22265  TopOnctopon 22282   CnP ccnp 22599  Filcfil 23219   fLim cflim 23308   fLimf cflf 23309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-ntr 22394  df-nei 22472  df-cnp 22602  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314
This theorem is referenced by:  cnpflf2  23374  cnpflf  23375  flfcnp  23378  cnpfcfi  23414
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