Theorem iscnp3 23222

Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.)

Assertion

Ref	Expression
iscnp3	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑃,𝑦

Proof of Theorem iscnp3

Step	Hyp	Ref	Expression
1		iscnp 23215	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
2		ffun 6666	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
3	2	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → Fun 𝐹)
4		toponss 22905	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
5	4	adantlr 716	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
6		fdm 6672	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
7	6	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → dom 𝐹 = 𝑋)
8	5, 7	sseqtrrd 3960	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ dom 𝐹)
9		funimass3 7001	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ⊆ dom 𝐹) → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
10	3, 8, 9	syl2anc 585	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
11	10	anbi2d 631	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
12	11	rexbidva 3160	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
13	12	imbi2d 340	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))
14	13	ralbidv 3161	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))
15	14	pm5.32da 579	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
16	15	3ad2ant1 1134	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
17	1, 16	bitrd 279	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Fun wfun 6487  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  TopOnctopon 22888   CnP ccnp 23203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-top 22872  df-topon 22889  df-cnp 23206

This theorem is referenced by:  cncnpi  23256  cnpdis  23271