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Theorem cnpimaex 21294
Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
Assertion
Ref Expression
cnpimaex ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝐾 ∧ (𝐹𝑃) ∈ 𝐴) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑃

Proof of Theorem cnpimaex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2817 . . . . . 6 𝐽 = 𝐽
2 eqid 2817 . . . . . 6 𝐾 = 𝐾
31, 2iscnp2 21277 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽) ∧ (𝐹: 𝐽 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
43simprbi 486 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹: 𝐽 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
54simprd 485 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
6 eleq2 2885 . . . . 5 (𝑦 = 𝐴 → ((𝐹𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝐴))
7 sseq2 3835 . . . . . . 7 (𝑦 = 𝐴 → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝐴))
87anbi2d 616 . . . . . 6 (𝑦 = 𝐴 → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴)))
98rexbidv 3251 . . . . 5 (𝑦 = 𝐴 → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴)))
106, 9imbi12d 335 . . . 4 (𝑦 = 𝐴 → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ((𝐹𝑃) ∈ 𝐴 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))))
1110rspccv 3510 . . 3 (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐴𝐾 → ((𝐹𝑃) ∈ 𝐴 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))))
125, 11syl 17 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐴𝐾 → ((𝐹𝑃) ∈ 𝐴 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))))
13123imp 1130 1 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝐾 ∧ (𝐹𝑃) ∈ 𝐴) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3107  wrex 3108  wss 3780   cuni 4641  cima 5327  wf 6106  cfv 6110  (class class class)co 6883  Topctop 20931   CnP ccnp 21263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-fv 6118  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-1st 7407  df-2nd 7408  df-map 8103  df-top 20932  df-topon 20949  df-cnp 21266
This theorem is referenced by:  iscnp4  21301  cnpnei  21302  cnpco  21305  cncnp  21318  cnpresti  21326  lmcnp  21342  txcnpi  21645  txcnp  21657  ptcnplem  21658  cnpflfi  22036  ghmcnp  22151  xrlimcnp  24931  cnambfre  33788
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