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Theorem iscnp4 21445
Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃 " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
iscnp4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑃,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Proof of Theorem iscnp4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cnpf2 21432 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
213expa 1151 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
323adantl3 1213 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
4 simplr 785 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
5 simpll2 1275 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐾 ∈ (TopOn‘𝑌))
6 topontop 21095 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
75, 6syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐾 ∈ Top)
8 eqid 2825 . . . . . . . . . 10 𝐾 = 𝐾
98neii1 21288 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 𝐾)
107, 9sylancom 582 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 𝐾)
118ntropn 21231 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
127, 10, 11syl2anc 579 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
13 simpr 479 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
143adantr 474 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐹:𝑋𝑌)
15 simpll3 1277 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑃𝑋)
1614, 15ffvelrnd 6614 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ 𝑌)
17 toponuni 21096 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
185, 17syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑌 = 𝐾)
1916, 18eleqtrd 2908 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ 𝐾)
2019snssd 4560 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → {(𝐹𝑃)} ⊆ 𝐾)
218neiint 21286 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ {(𝐹𝑃)} ⊆ 𝐾𝑦 𝐾) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) ↔ {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
227, 20, 10, 21syl3anc 1494 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) ↔ {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
2313, 22mpbid 224 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦))
24 fvex 6450 . . . . . . . . 9 (𝐹𝑃) ∈ V
2524snss 4537 . . . . . . . 8 ((𝐹𝑃) ∈ ((int‘𝐾)‘𝑦) ↔ {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦))
2623, 25sylibr 226 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ ((int‘𝐾)‘𝑦))
27 cnpimaex 21438 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ ((int‘𝐾)‘𝑦) ∈ 𝐾 ∧ (𝐹𝑃) ∈ ((int‘𝐾)‘𝑦)) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))
284, 12, 26, 27syl3anc 1494 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))
29 simpl1 1246 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
3029ad2antrr 717 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝐽 ∈ (TopOn‘𝑋))
31 topontop 21095 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3230, 31syl 17 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝐽 ∈ Top)
33 simprl 787 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑥𝐽)
34 simprrl 799 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑃𝑥)
35 opnneip 21301 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑃𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
3632, 33, 34, 35syl3anc 1494 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
37 simprrr 800 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦))
388ntrss2 21239 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
397, 10, 38syl2anc 579 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
4039adantr 474 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
4137, 40sstrd 3837 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → (𝐹𝑥) ⊆ 𝑦)
4228, 36, 41reximssdv 3227 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)
4342ralrimiva 3175 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)
443, 43jca 507 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦))
4544ex 403 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)))
46 simpll2 1275 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝐾 ∈ (TopOn‘𝑌))
4746, 6syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝐾 ∈ Top)
48 simprl 787 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑦𝐾)
49 simprr 789 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → (𝐹𝑃) ∈ 𝑦)
50 opnneip 21301 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
5147, 48, 49, 50syl3anc 1494 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
52 simpl1 1246 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
5352ad2antrr 717 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
5453, 31syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝐽 ∈ Top)
55 simprl 787 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
56 eqid 2825 . . . . . . . . . . . . . 14 𝐽 = 𝐽
5756neii1 21288 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑥 𝐽)
5854, 55, 57syl2anc 579 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑥 𝐽)
5956ntropn 21231 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
6054, 58, 59syl2anc 579 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
61 simpll3 1277 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑃𝑋)
6261adantr 474 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃𝑋)
63 toponuni 21096 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
6453, 63syl 17 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑋 = 𝐽)
6562, 64eleqtrd 2908 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃 𝐽)
6665snssd 4560 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → {𝑃} ⊆ 𝐽)
6756neiint 21286 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝐽𝑥 𝐽) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
6854, 66, 58, 67syl3anc 1494 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
6955, 68mpbid 224 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → {𝑃} ⊆ ((int‘𝐽)‘𝑥))
70 snssg 4536 . . . . . . . . . . . . 13 (𝑃𝑋 → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
7162, 70syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
7269, 71mpbird 249 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃 ∈ ((int‘𝐽)‘𝑥))
7356ntrss2 21239 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
7454, 58, 73syl2anc 579 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
75 imass2 5746 . . . . . . . . . . . . 13 (((int‘𝐽)‘𝑥) ⊆ 𝑥 → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹𝑥))
7674, 75syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹𝑥))
77 simprr 789 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐹𝑥) ⊆ 𝑦)
7876, 77sstrd 3837 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)
79 eleq2 2895 . . . . . . . . . . . . 13 (𝑧 = ((int‘𝐽)‘𝑥) → (𝑃𝑧𝑃 ∈ ((int‘𝐽)‘𝑥)))
80 imaeq2 5707 . . . . . . . . . . . . . 14 (𝑧 = ((int‘𝐽)‘𝑥) → (𝐹𝑧) = (𝐹 “ ((int‘𝐽)‘𝑥)))
8180sseq1d 3857 . . . . . . . . . . . . 13 (𝑧 = ((int‘𝐽)‘𝑥) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦))
8279, 81anbi12d 624 . . . . . . . . . . . 12 (𝑧 = ((int‘𝐽)‘𝑥) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)))
8382rspcev 3526 . . . . . . . . . . 11 ((((int‘𝐽)‘𝑥) ∈ 𝐽 ∧ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
8460, 72, 78, 83syl12anc 870 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
8584rexlimdvaa 3241 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → (∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
8651, 85embantd 59 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
8786ex 403 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))))
8887com23 86 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → ((𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))))
8988exp4a 424 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → (𝑦𝐾 → ((𝐹𝑃) ∈ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))))
9089ralimdv2 3170 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦 → ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))))
9190imdistanda 567 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))))
92 iscnp 21419 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))))
9391, 92sylibrd 251 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)))
9445, 93impbid 204 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  wss 3798  {csn 4399   cuni 4660  cima 5349  wf 6123  cfv 6127  (class class class)co 6910  Topctop 21075  TopOnctopon 21092  intcnt 21199  neicnei 21279   CnP ccnp 21407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-top 21076  df-topon 21093  df-ntr 21202  df-nei 21280  df-cnp 21410
This theorem is referenced by:  cnnei  21464
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