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Theorem lmss 12342
Description: Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
Hypotheses
Ref Expression
lmss.1 𝐾 = (𝐽t 𝑌)
lmss.2 𝑍 = (ℤ𝑀)
lmss.3 (𝜑𝑌𝑉)
lmss.4 (𝜑𝐽 ∈ Top)
lmss.5 (𝜑𝑃𝑌)
lmss.6 (𝜑𝑀 ∈ ℤ)
lmss.7 (𝜑𝐹:𝑍𝑌)
Assertion
Ref Expression
lmss (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))

Proof of Theorem lmss
Dummy variables 𝑗 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmss.4 . . . . . 6 (𝜑𝐽 ∈ Top)
2 eqid 2117 . . . . . . 7 𝐽 = 𝐽
32toptopon 12112 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 121 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
5 lmcl 12341 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
64, 5sylan 281 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
7 lmfss 12340 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
84, 7sylan 281 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
9 rnss 4739 . . . . . 6 (𝐹 ⊆ (ℂ × 𝐽) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
108, 9syl 14 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
11 rnxpss 4940 . . . . 5 ran (ℂ × 𝐽) ⊆ 𝐽
1210, 11sstrdi 3079 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 𝐽)
136, 12jca 304 . . 3 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
1413ex 114 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
15 inss2 3267 . . . . 5 (𝑌 𝐽) ⊆ 𝐽
16 lmss.1 . . . . . . 7 𝐾 = (𝐽t 𝑌)
17 lmss.3 . . . . . . . 8 (𝜑𝑌𝑉)
18 resttopon2 12274 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑌𝑉) → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
194, 17, 18syl2anc 408 . . . . . . 7 (𝜑 → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
2016, 19eqeltrid 2204 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑌 𝐽)))
21 lmcl 12341 . . . . . 6 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2220, 21sylan 281 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2315, 22sseldi 3065 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 𝐽)
24 lmfss 12340 . . . . . . . 8 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
2520, 24sylan 281 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
26 rnss 4739 . . . . . . 7 (𝐹 ⊆ (ℂ × (𝑌 𝐽)) → ran 𝐹 ⊆ ran (ℂ × (𝑌 𝐽)))
2725, 26syl 14 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 ⊆ ran (ℂ × (𝑌 𝐽)))
28 rnxpss 4940 . . . . . 6 ran (ℂ × (𝑌 𝐽)) ⊆ (𝑌 𝐽)
2927, 28sstrdi 3079 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 ⊆ (𝑌 𝐽))
3029, 15sstrdi 3079 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 𝐽)
3123, 30jca 304 . . 3 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
3231ex 114 . 2 (𝜑 → (𝐹(⇝𝑡𝐾)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
33 simprl 505 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 𝐽)
34 lmss.5 . . . . . . . 8 (𝜑𝑃𝑌)
3534adantr 274 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃𝑌)
3635, 33elind 3231 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 ∈ (𝑌 𝐽))
3733, 362thd 174 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃 𝐽𝑃 ∈ (𝑌 𝐽)))
3816eleq2i 2184 . . . . . . . . 9 (𝑣𝐾𝑣 ∈ (𝐽t 𝑌))
391adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ Top)
4017adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑌𝑉)
41 elrest 12054 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4239, 40, 41syl2anc 408 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4342biimpa 294 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣 ∈ (𝐽t 𝑌)) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
4438, 43sylan2b 285 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
45 r19.29r 2547 . . . . . . . . . 10 ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → ∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
4635biantrud 302 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢 ↔ (𝑃𝑢𝑃𝑌)))
47 elin 3229 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ (𝑢𝑌) ↔ (𝑃𝑢𝑃𝑌))
4846, 47syl6bbr 197 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢𝑃 ∈ (𝑢𝑌)))
49 lmss.2 . . . . . . . . . . . . . . . . . . . . 21 𝑍 = (ℤ𝑀)
5049uztrn2 9311 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
51 lmss.7 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹:𝑍𝑌)
5251adantr 274 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍𝑌)
5352ffvelrnda 5523 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) ∈ 𝑌)
5453biantrud 302 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌)))
55 elin 3229 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ (𝑢𝑌) ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌))
5654, 55syl6bbr 197 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5750, 56sylan2 284 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5857anassrs 397 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5958ralbidva 2410 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6059rexbidva 2411 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6148, 60imbi12d 233 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6261adantr 274 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6362biimpd 143 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
64 eleq2 2181 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (𝑃𝑣𝑃 ∈ (𝑢𝑌)))
65 eleq2 2181 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑢𝑌) → ((𝐹𝑘) ∈ 𝑣 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
6665rexralbidv 2438 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6764, 66imbi12d 233 . . . . . . . . . . . . . 14 (𝑣 = (𝑢𝑌) → ((𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6867imbi2d 229 . . . . . . . . . . . . 13 (𝑣 = (𝑢𝑌) → (((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)) ↔ ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))))
6963, 68syl5ibrcom 156 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑣 = (𝑢𝑌) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
7069impd 252 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7170rexlimdva 2526 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7245, 71syl5 32 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7372expdimp 257 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ ∃𝑢𝐽 𝑣 = (𝑢𝑌)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7444, 73syldan 280 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7574ralrimdva 2489 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7639adantr 274 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝐽 ∈ Top)
7740adantr 274 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑌𝑉)
78 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑢𝐽)
79 elrestr 12055 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8076, 77, 78, 79syl3anc 1201 . . . . . . . . . 10 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8180, 16eleqtrrdi 2211 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ 𝐾)
8267rspcv 2759 . . . . . . . . 9 ((𝑢𝑌) ∈ 𝐾 → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
8381, 82syl 14 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
8483, 62sylibrd 168 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8584ralrimdva 2489 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8675, 85impbid 128 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
8737, 86anbi12d 464 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) ↔ (𝑃 ∈ (𝑌 𝐽) ∧ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
8839, 3sylib 121 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐽))
89 lmss.6 . . . . . 6 (𝜑𝑀 ∈ ℤ)
9089adantr 274 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑀 ∈ ℤ)
9152ffnd 5243 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹 Fn 𝑍)
92 simprr 506 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 𝐽)
93 df-f 5097 . . . . . 6 (𝐹:𝑍 𝐽 ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 𝐽))
9491, 92, 93sylanbrc 413 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍 𝐽)
95 eqidd 2118 . . . . 5 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
9688, 49, 90, 94, 95lmbrf 12311 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))))
9720adantr 274 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐾 ∈ (TopOn‘(𝑌 𝐽)))
9852frnd 5252 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹𝑌)
9998, 92ssind 3270 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 ⊆ (𝑌 𝐽))
100 df-f 5097 . . . . . 6 (𝐹:𝑍⟶(𝑌 𝐽) ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 ⊆ (𝑌 𝐽)))
10191, 99, 100sylanbrc 413 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍⟶(𝑌 𝐽))
10297, 49, 90, 101, 95lmbrf 12311 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐾)𝑃 ↔ (𝑃 ∈ (𝑌 𝐽) ∧ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
10387, 96, 1023bitr4d 219 . . 3 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
104103ex 114 . 2 (𝜑 → ((𝑃 𝐽 ∧ ran 𝐹 𝐽) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃)))
10514, 32, 104pm5.21ndd 679 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wral 2393  wrex 2394  cin 3040  wss 3041   cuni 3706   class class class wbr 3899   × cxp 4507  ran crn 4510   Fn wfn 5088  wf 5089  cfv 5093  (class class class)co 5742  cc 7586  cz 9022  cuz 9294  t crest 12047  Topctop 12091  TopOnctopon 12104  𝑡clm 12283
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pm 6513  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-rest 12049  df-topgen 12068  df-top 12092  df-topon 12105  df-bases 12137  df-lm 12286
This theorem is referenced by: (None)
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