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Theorem cnptoprest2 12441
 Description: Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
Hypotheses
Ref Expression
cnprest.1 𝑋 = 𝐽
cnprest.2 𝑌 = 𝐾
Assertion
Ref Expression
cnptoprest2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))

Proof of Theorem cnptoprest2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnprest.1 . . . . . . . 8 𝑋 = 𝐽
21toptopon 12217 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
32biimpi 119 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
43ad2antrr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
54adantr 274 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
6 simplr 520 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐾 ∈ Top)
76adantr 274 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
8 simpr 109 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
9 cnprcl2k 12407 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
105, 7, 8, 9syl3anc 1217 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
1110ex 114 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋))
124adantr 274 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
13 cnprest.2 . . . . . . . . 9 𝑌 = 𝐾
14 uniexg 4367 . . . . . . . . 9 (𝐾 ∈ Top → 𝐾 ∈ V)
1513, 14eqeltrid 2227 . . . . . . . 8 (𝐾 ∈ Top → 𝑌 ∈ V)
166, 15syl 14 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝑌 ∈ V)
17 simprr 522 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐵𝑌)
1816, 17ssexd 4074 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐵 ∈ V)
19 resttop 12371 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝐾t 𝐵) ∈ Top)
206, 18, 19syl2anc 409 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐾t 𝐵) ∈ Top)
2120adantr 274 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → (𝐾t 𝐵) ∈ Top)
22 simpr 109 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃))
23 cnprcl2k 12407 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾t 𝐵) ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝑃𝑋)
2412, 21, 22, 23syl3anc 1217 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝑃𝑋)
2524ex 114 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃) → 𝑃𝑋))
26 simprl 521 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐹:𝑋𝐵)
2726ffvelrnda 5561 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹𝑃) ∈ 𝐵)
2827biantrud 302 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑃) ∈ 𝑥 ↔ ((𝐹𝑃) ∈ 𝑥 ∧ (𝐹𝑃) ∈ 𝐵)))
29 elin 3262 . . . . . . . 8 ((𝐹𝑃) ∈ (𝑥𝐵) ↔ ((𝐹𝑃) ∈ 𝑥 ∧ (𝐹𝑃) ∈ 𝐵))
3028, 29syl6bbr 197 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑃) ∈ 𝑥 ↔ (𝐹𝑃) ∈ (𝑥𝐵)))
31 imassrn 4898 . . . . . . . . . . . 12 (𝐹𝑦) ⊆ ran 𝐹
32 simplrl 525 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐹:𝑋𝐵)
3332frnd 5288 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ran 𝐹𝐵)
3431, 33sstrid 3111 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹𝑦) ⊆ 𝐵)
3534biantrud 302 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑦) ⊆ 𝑥 ↔ ((𝐹𝑦) ⊆ 𝑥 ∧ (𝐹𝑦) ⊆ 𝐵)))
36 ssin 3301 . . . . . . . . . 10 (((𝐹𝑦) ⊆ 𝑥 ∧ (𝐹𝑦) ⊆ 𝐵) ↔ (𝐹𝑦) ⊆ (𝑥𝐵))
3735, 36syl6bb 195 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑦) ⊆ 𝑥 ↔ (𝐹𝑦) ⊆ (𝑥𝐵)))
3837anbi2d 460 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥) ↔ (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
3938rexbidv 2439 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥) ↔ ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
4030, 39imbi12d 233 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
4140ralbidv 2438 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ ∀𝑥𝐾 ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
42 vex 2692 . . . . . . . 8 𝑥 ∈ V
4342inex1 4068 . . . . . . 7 (𝑥𝐵) ∈ V
4443a1i 9 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) ∧ 𝑥𝐾) → (𝑥𝐵) ∈ V)
456adantr 274 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐾 ∈ Top)
4618adantr 274 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐵 ∈ V)
47 elrest 12159 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑧 = (𝑥𝐵)))
4845, 46, 47syl2anc 409 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝑧 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑧 = (𝑥𝐵)))
49 eleq2 2204 . . . . . . . 8 (𝑧 = (𝑥𝐵) → ((𝐹𝑃) ∈ 𝑧 ↔ (𝐹𝑃) ∈ (𝑥𝐵)))
50 sseq2 3124 . . . . . . . . . 10 (𝑧 = (𝑥𝐵) → ((𝐹𝑦) ⊆ 𝑧 ↔ (𝐹𝑦) ⊆ (𝑥𝐵)))
5150anbi2d 460 . . . . . . . . 9 (𝑧 = (𝑥𝐵) → ((𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧) ↔ (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
5251rexbidv 2439 . . . . . . . 8 (𝑧 = (𝑥𝐵) → (∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧) ↔ ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
5349, 52imbi12d 233 . . . . . . 7 (𝑧 = (𝑥𝐵) → (((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
5453adantl 275 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) ∧ 𝑧 = (𝑥𝐵)) → (((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
5544, 48, 54ralxfr2d 4391 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ ∀𝑥𝐾 ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
5641, 55bitr4d 190 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧))))
574adantr 274 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5813toptopon 12217 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
5945, 58sylib 121 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐾 ∈ (TopOn‘𝑌))
60 simpr 109 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝑃𝑋)
61 iscnp 12400 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))))
6257, 59, 60, 61syl3anc 1217 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))))
6317adantr 274 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐵𝑌)
6432, 63fssd 5291 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐹:𝑋𝑌)
6564biantrurd 303 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))))
6662, 65bitr4d 190 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))))
67 resttopon 12372 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
6859, 63, 67syl2anc 409 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
69 iscnp 12400 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃) ↔ (𝐹:𝑋𝐵 ∧ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)))))
7057, 68, 60, 69syl3anc 1217 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃) ↔ (𝐹:𝑋𝐵 ∧ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)))))
7126biantrurd 303 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋𝐵 ∧ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)))))
7271adantr 274 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋𝐵 ∧ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)))))
7370, 72bitr4d 190 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃) ↔ ∀𝑧 ∈ (𝐾t 𝐵)((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧))))
7456, 66, 733bitr4d 219 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))
7574ex 114 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝑃𝑋 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃))))
7611, 25, 75pm5.21ndd 695 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ∩ cin 3073   ⊆ wss 3074  ∪ cuni 3742  ran crn 4546   “ cima 4548  ⟶wf 5125  ‘cfv 5129  (class class class)co 5780   ↾t crest 12152  Topctop 12196  TopOnctopon 12209   CnP ccnp 12387 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-map 6550  df-rest 12154  df-topgen 12173  df-top 12197  df-topon 12210  df-bases 12242  df-cnp 12390 This theorem is referenced by: (None)
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