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Theorem icnpimaex 13982
Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
Assertion
Ref Expression
icnpimaex (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝑃   π‘₯,𝑋   π‘₯,π‘Œ

Proof of Theorem icnpimaex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr3 1006 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ (πΉβ€˜π‘ƒ) ∈ 𝐴)
2 eleq2 2251 . . . 4 (𝑦 = 𝐴 β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 ↔ (πΉβ€˜π‘ƒ) ∈ 𝐴))
3 sseq2 3191 . . . . . 6 (𝑦 = 𝐴 β†’ ((𝐹 β€œ π‘₯) βŠ† 𝑦 ↔ (𝐹 β€œ π‘₯) βŠ† 𝐴))
43anbi2d 464 . . . . 5 (𝑦 = 𝐴 β†’ ((𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) ↔ (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴)))
54rexbidv 2488 . . . 4 (𝑦 = 𝐴 β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴)))
62, 5imbi12d 234 . . 3 (𝑦 = 𝐴 β†’ (((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)) ↔ ((πΉβ€˜π‘ƒ) ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴))))
7 simpr1 1004 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))
8 iscnp 13970 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
98adantr 276 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))))
107, 9mpbid 147 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))))
1110simprd 114 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ βˆ€π‘¦ ∈ 𝐾 ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
12 simpr2 1005 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ 𝐴 ∈ 𝐾)
136, 11, 12rspcdva 2858 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴)))
141, 13mpd 13 1 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 979   = wceq 1363   ∈ wcel 2158  βˆ€wral 2465  βˆƒwrex 2466   βŠ† wss 3141   β€œ cima 4641  βŸΆwf 5224  β€˜cfv 5228  (class class class)co 5888  TopOnctopon 13781   CnP ccnp 13957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-top 13769  df-topon 13782  df-cnp 13960
This theorem is referenced by:  iscnp4  13989  cnpnei  13990  cnptopco  13993  cncnp  14001  cnptopresti  14009  lmtopcnp  14021  txcnp  14042
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