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Theorem iscnp4 20877
Description: The predicate "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 20864 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
213expa 1257 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
323adantl3 1212 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
4 simplr 788 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
5 simpll2 1094 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐾 ∈ (TopOn‘𝑌))
6 topontop 20541 . . . . . . . . 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 2610 . . . . . . . . . 10 𝐾 = 𝐾
98neii1 20720 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 𝐾)
107, 9sylancom 698 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 𝐾)
118ntropn 20663 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
127, 10, 11syl2anc 691 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
13 simpr 476 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
143adantr 480 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝐹:𝑋𝑌)
15 simpll3 1095 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → 𝑃𝑋)
1614, 15ffvelrnd 6268 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ 𝑌)
17 toponuni 20542 . . . . . . . . . . . . 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 2690 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ 𝐾)
2019snssd 4281 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → {(𝐹𝑃)} ⊆ 𝐾)
218neiint 20718 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ {(𝐹𝑃)} ⊆ 𝐾𝑦 𝐾) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}) ↔ {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
227, 20, 10, 21syl3anc 1318 . . . . . . . . 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 221 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦))
24 fvex 6113 . . . . . . . . 9 (𝐹𝑃) ∈ V
2524snss 4259 . . . . . . . 8 ((𝐹𝑃) ∈ ((int‘𝐾)‘𝑦) ↔ {(𝐹𝑃)} ⊆ ((int‘𝐾)‘𝑦))
2623, 25sylibr 223 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → (𝐹𝑃) ∈ ((int‘𝐾)‘𝑦))
27 cnpimaex 20870 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ ((int‘𝐾)‘𝑦) ∈ 𝐾 ∧ (𝐹𝑃) ∈ ((int‘𝐾)‘𝑦)) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))
284, 12, 26, 27syl3anc 1318 . . . . . 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 1057 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
3029ad2antrr 758 . . . . . . . . . . 11 (((((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 20541 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3230, 31syl 17 . . . . . . . . . 10 (((((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 790 . . . . . . . . . 10 (((((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 800 . . . . . . . . . 10 (((((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 20733 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑃𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
3632, 33, 34, 35syl3anc 1318 . . . . . . . . 9 (((((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 801 . . . . . . . . . 10 (((((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 20671 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
397, 10, 38syl2anc 691 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
4039adantr 480 . . . . . . . . . 10 (((((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 3578 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) ∧ (𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → (𝐹𝑥) ⊆ 𝑦)
4236, 41jca 553 . . . . . . . 8 (((((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}) ∧ (𝐹𝑥) ⊆ 𝑦))
4342ex 449 . . . . . . 7 ((((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}) ∧ (𝐹𝑥) ⊆ 𝑦)))
4443reximdv2 2997 . . . . . 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})(𝐹𝑥) ⊆ 𝑦))
4528, 44mpd 15 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)
4645ralrimiva 2949 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)
473, 46jca 553 . . 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))
4847ex 449 . 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)))
49 simpll2 1094 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝐾 ∈ (TopOn‘𝑌))
5049, 6syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝐾 ∈ Top)
51 simprl 790 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑦𝐾)
52 simprr 792 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → (𝐹𝑃) ∈ 𝑦)
53 opnneip 20733 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
5450, 51, 52, 53syl3anc 1318 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)}))
55 simpl1 1057 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
5655ad2antrr 758 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
5756, 31syl 17 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝐽 ∈ Top)
58 simprl 790 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
59 eqid 2610 . . . . . . . . . . . . . 14 𝐽 = 𝐽
6059neii1 20720 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑥 𝐽)
6157, 58, 60syl2anc 691 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑥 𝐽)
6259ntropn 20663 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
6357, 61, 62syl2anc 691 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
64 simpll3 1095 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → 𝑃𝑋)
6564adantr 480 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃𝑋)
66 toponuni 20542 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
6756, 66syl 17 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑋 = 𝐽)
6865, 67eleqtrd 2690 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃 𝐽)
6968snssd 4281 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → {𝑃} ⊆ 𝐽)
7059neiint 20718 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝐽𝑥 𝐽) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
7157, 69, 61, 70syl3anc 1318 . . . . . . . . . . . . 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)))
7258, 71mpbid 221 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → {𝑃} ⊆ ((int‘𝐽)‘𝑥))
73 snssg 4268 . . . . . . . . . . . . 13 (𝑃𝑋 → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
7465, 73syl 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)))
7572, 74mpbird 246 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → 𝑃 ∈ ((int‘𝐽)‘𝑥))
7659ntrss2 20671 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
7757, 61, 76syl2anc 691 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
78 imass2 5420 . . . . . . . . . . . . 13 (((int‘𝐽)‘𝑥) ⊆ 𝑥 → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹𝑥))
7977, 78syl 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))
80 simprr 792 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐹𝑥) ⊆ 𝑦)
8179, 80sstrd 3578 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)
82 eleq2 2677 . . . . . . . . . . . . 13 (𝑧 = ((int‘𝐽)‘𝑥) → (𝑃𝑧𝑃 ∈ ((int‘𝐽)‘𝑥)))
83 imaeq2 5381 . . . . . . . . . . . . . 14 (𝑧 = ((int‘𝐽)‘𝑥) → (𝐹𝑧) = (𝐹 “ ((int‘𝐽)‘𝑥)))
8483sseq1d 3595 . . . . . . . . . . . . 13 (𝑧 = ((int‘𝐽)‘𝑥) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦))
8582, 84anbi12d 743 . . . . . . . . . . . 12 (𝑧 = ((int‘𝐽)‘𝑥) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)))
8685rspcev 3282 . . . . . . . . . . 11 ((((int‘𝐽)‘𝑥) ∈ 𝐽 ∧ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
8763, 75, 81, 86syl12anc 1316 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹𝑥) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
8887rexlimdvaa 3014 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → (∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
8954, 88embantd 57 . . . . . . . 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)))
9089ex 449 . . . . . . 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))))
9190com23 84 . . . . . 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))))
9291exp4a 631 . . . . 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)))))
9392ralimdv2 2944 . . . 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))))
9493imdistanda 725 . . 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)))))
95 iscnp 20851 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))))
9694, 95sylibrd 248 . 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)))
9748, 96impbid 201 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540  {csn 4125   cuni 4372  cima 5041  wf 5800  cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518  intcnt 20631  neicnei 20711   CnP ccnp 20839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-ntr 20634  df-nei 20712  df-cnp 20842
This theorem is referenced by:  cnnei  20896
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