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Theorem cnptoprest2 14011
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 13789 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
32biimpi 120 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
43ad2antrr 488 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
54adantr 276 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
6 simplr 528 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐾 ∈ Top)
76adantr 276 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
8 simpr 110 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
9 cnprcl2k 13977 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
105, 7, 8, 9syl3anc 1248 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
1110ex 115 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋))
124adantr 276 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
13 cnprest.2 . . . . . . . . 9 𝑌 = 𝐾
14 uniexg 4451 . . . . . . . . 9 (𝐾 ∈ Top → 𝐾 ∈ V)
1513, 14eqeltrid 2274 . . . . . . . 8 (𝐾 ∈ Top → 𝑌 ∈ V)
166, 15syl 14 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝑌 ∈ V)
17 simprr 531 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐵𝑌)
1816, 17ssexd 4155 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐵 ∈ V)
19 resttop 13941 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝐾t 𝐵) ∈ Top)
206, 18, 19syl2anc 411 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐾t 𝐵) ∈ Top)
2120adantr 276 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → (𝐾t 𝐵) ∈ Top)
22 simpr 110 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃))
23 cnprcl2k 13977 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾t 𝐵) ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝑃𝑋)
2412, 21, 22, 23syl3anc 1248 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)) → 𝑃𝑋)
2524ex 115 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃) → 𝑃𝑋))
26 simprl 529 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → 𝐹:𝑋𝐵)
2726ffvelcdmda 5664 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹𝑃) ∈ 𝐵)
2827biantrud 304 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑃) ∈ 𝑥 ↔ ((𝐹𝑃) ∈ 𝑥 ∧ (𝐹𝑃) ∈ 𝐵)))
29 elin 3330 . . . . . . . 8 ((𝐹𝑃) ∈ (𝑥𝐵) ↔ ((𝐹𝑃) ∈ 𝑥 ∧ (𝐹𝑃) ∈ 𝐵))
3028, 29bitr4di 198 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑃) ∈ 𝑥 ↔ (𝐹𝑃) ∈ (𝑥𝐵)))
31 imassrn 4993 . . . . . . . . . . . 12 (𝐹𝑦) ⊆ ran 𝐹
32 simplrl 535 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐹:𝑋𝐵)
3332frnd 5387 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ran 𝐹𝐵)
3431, 33sstrid 3178 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹𝑦) ⊆ 𝐵)
3534biantrud 304 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑦) ⊆ 𝑥 ↔ ((𝐹𝑦) ⊆ 𝑥 ∧ (𝐹𝑦) ⊆ 𝐵)))
36 ssin 3369 . . . . . . . . . 10 (((𝐹𝑦) ⊆ 𝑥 ∧ (𝐹𝑦) ⊆ 𝐵) ↔ (𝐹𝑦) ⊆ (𝑥𝐵))
3735, 36bitrdi 196 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝐹𝑦) ⊆ 𝑥 ↔ (𝐹𝑦) ⊆ (𝑥𝐵)))
3837anbi2d 464 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → ((𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥) ↔ (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
3938rexbidv 2488 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥) ↔ ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
4030, 39imbi12d 234 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
4140ralbidv 2487 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)) ↔ ∀𝑥𝐾 ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
42 vex 2752 . . . . . . . 8 𝑥 ∈ V
4342inex1 4149 . . . . . . 7 (𝑥𝐵) ∈ V
4443a1i 9 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) ∧ 𝑥𝐾) → (𝑥𝐵) ∈ V)
456adantr 276 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐾 ∈ Top)
4618adantr 276 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐵 ∈ V)
47 elrest 12712 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑧 = (𝑥𝐵)))
4845, 46, 47syl2anc 411 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝑧 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑧 = (𝑥𝐵)))
49 eleq2 2251 . . . . . . . 8 (𝑧 = (𝑥𝐵) → ((𝐹𝑃) ∈ 𝑧 ↔ (𝐹𝑃) ∈ (𝑥𝐵)))
50 sseq2 3191 . . . . . . . . . 10 (𝑧 = (𝑥𝐵) → ((𝐹𝑦) ⊆ 𝑧 ↔ (𝐹𝑦) ⊆ (𝑥𝐵)))
5150anbi2d 464 . . . . . . . . 9 (𝑧 = (𝑥𝐵) → ((𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧) ↔ (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
5251rexbidv 2488 . . . . . . . 8 (𝑧 = (𝑥𝐵) → (∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧) ↔ ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵))))
5349, 52imbi12d 234 . . . . . . 7 (𝑧 = (𝑥𝐵) → (((𝐹𝑃) ∈ 𝑧 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑧)) ↔ ((𝐹𝑃) ∈ (𝑥𝐵) → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ (𝑥𝐵)))))
5453adantl 277 . . . . . 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 4476 . . . . 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 191 . . . 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 276 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5813toptopon 13789 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
5945, 58sylib 122 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐾 ∈ (TopOn‘𝑌))
60 simpr 110 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝑃𝑋)
61 iscnp 13970 . . . . . 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 1248 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥)))))
6317adantr 276 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐵𝑌)
6432, 63fssd 5390 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → 𝐹:𝑋𝑌)
6564biantrurd 305 . . . . 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 191 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑥𝐾 ((𝐹𝑃) ∈ 𝑥 → ∃𝑦𝐽 (𝑃𝑦 ∧ (𝐹𝑦) ⊆ 𝑥))))
67 resttopon 13942 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
6859, 63, 67syl2anc 411 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
69 iscnp 13970 . . . . . 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 1248 . . . . 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 305 . . . . . 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 276 . . . . 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 191 . . . 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 220 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))
7574ex 115 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝐵𝐵𝑌)) → (𝑃𝑋 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃))))
7611, 25, 75pm5.21ndd 706 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 104  wb 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  Vcvv 2749  cin 3140  wss 3141   cuni 3821  ran crn 4639  cima 4641  wf 5224  cfv 5228  (class class class)co 5888  t crest 12705  Topctop 13768  TopOnctopon 13781   CnP ccnp 13957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-rest 12707  df-topgen 12726  df-top 13769  df-topon 13782  df-bases 13814  df-cnp 13960
This theorem is referenced by: (None)
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