Theorem cnpimaex 21282

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 2760	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2760	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	iscnp2 21265	. . . . 5 ⊢ (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 483	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))
5	4	simprd 482	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
6		eleq2 2828	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑃) ∈ 𝑦 ↔ (𝐹‘𝑃) ∈ 𝐴))
7		sseq2 3768	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
8	7	anbi2d 742	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)))
9	8	rexbidv 3190	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)))
10	6, 9	imbi12d 333	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ((𝐹‘𝑃) ∈ 𝐴 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))))
11	10	rspccv 3446	. . 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))))
12	5, 11	syl 17	. 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐴 ∈ 𝐾 → ((𝐹‘𝑃) ∈ 𝐴 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))))
13	12	3imp 1102	1 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ⊆ wss 3715  ∪ cuni 4588   “ cima 5269  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814  Topctop 20920   CnP ccnp 21251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-top 20921  df-topon 20938  df-cnp 21254

This theorem is referenced by:  iscnp4  21289  cnpnei  21290  cnpco  21293  cncnp  21306  cnpresti  21314  lmcnp  21330  txcnpi  21633  txcnp  21645  ptcnplem  21646  cnpflfi  22024  ghmcnp  22139  xrlimcnp  24915  cnambfre  33789