Theorem cnpimaex 22760

Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Assertion

Ref	Expression
cnpimaex	⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 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.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2733	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	iscnp2 22743	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
4	3	simprbi 498	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))
5		eleq2 2823	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑃) ∈ 𝑦 ↔ (𝐹‘𝑃) ∈ 𝐴))
6		sseq2 4009	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
7	6	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)))
8	7	rexbidv 3179	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)))
9	5, 8	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ((𝐹‘𝑃) ∈ 𝐴 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))))
10	9	rspccv 3610	. . 3 ⊢ (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝐴 ∈ 𝐾 → ((𝐹‘𝑃) ∈ 𝐴 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))))
11	4, 10	simpl2im 505	. 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐴 ∈ 𝐾 → ((𝐹‘𝑃) ∈ 𝐴 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))))
12	11	3imp 1112	1 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949  ∪ cuni 4909   “ cima 5680  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Topctop 22395   CnP ccnp 22729

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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-top 22396  df-topon 22413  df-cnp 22732

This theorem is referenced by:  iscnp4  22767  cnpnei  22768  cnpco  22771  cncnp  22784  cnpresti  22792  lmcnp  22808  txcnpi  23112  txcnp  23124  ptcnplem  23125  cnpflfi  23503  ghmcnp  23619  xrlimcnp  26473  cnambfre  36536