Theorem lmtopcnp 14977

Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)

Hypotheses

Ref	Expression
lmcnp.3	⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)
lmcnp.k	⊢ (𝜑 → 𝐾 ∈ Top)
lmcnp.4	⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))

Assertion

Ref	Expression
lmtopcnp	⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))

Proof of Theorem lmtopcnp

Dummy variables 𝑗 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmcnp.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)
2		lmrcl 14919	. . . . . . . 8 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
4		toptopon2 14746	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
5	3, 4	sylib 122	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
6		lmcnp.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
7		toptopon2 14746	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
8	6, 7	sylib 122	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		lmcnp.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))
10		cnpf2 14934	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐺:∪ 𝐽⟶∪ 𝐾)
11	5, 8, 9, 10	syl3anc 1273	. . . . 5 ⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾)
12		nnuz 9792	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
13		1zzd 9506	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
14	5, 12, 13	lmbr2 14941	. . . . . . . . 9 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑃 ∈ ∪ 𝐽 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)))))
15	1, 14	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑃 ∈ ∪ 𝐽 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣))))
16	15	simp1d 1035	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝐽 ↑pm ℂ))
17		uniexg 4536	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
18	3, 17	syl 14	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
19		cnex 8156	. . . . . . . 8 ⊢ ℂ ∈ V
20		elpm2g 6834	. . . . . . . 8 ⊢ ((∪ 𝐽 ∈ V ∧ ℂ ∈ V) → (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶∪ 𝐽 ∧ dom 𝐹 ⊆ ℂ)))
21	18, 19, 20	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶∪ 𝐽 ∧ dom 𝐹 ⊆ ℂ)))
22	16, 21	mpbid 147	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶∪ 𝐽 ∧ dom 𝐹 ⊆ ℂ))
23	22	simpld 112	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶∪ 𝐽)
24		fco 5500	. . . . 5 ⊢ ((𝐺:∪ 𝐽⟶∪ 𝐾 ∧ 𝐹:dom 𝐹⟶∪ 𝐽) → (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾)
25	11, 23, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾)
26	25	fdmd 5489	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ 𝐹) = dom 𝐹)
27	26	feq2d 5470	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ↔ (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾))
28	25, 27	mpbird 167	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾)
29	22	simprd 114	. . . 4 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
30	26, 29	eqsstrd 3263	. . 3 ⊢ (𝜑 → dom (𝐺 ∘ 𝐹) ⊆ ℂ)
31		uniexg 4536	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
32	6, 31	syl 14	. . . 4 ⊢ (𝜑 → ∪ 𝐾 ∈ V)
33		elpm2g 6834	. . . 4 ⊢ ((∪ 𝐾 ∈ V ∧ ℂ ∈ V) → ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm ℂ) ↔ ((𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ∧ dom (𝐺 ∘ 𝐹) ⊆ ℂ)))
34	32, 19, 33	sylancl 413	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm ℂ) ↔ ((𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ∧ dom (𝐺 ∘ 𝐹) ⊆ ℂ)))
35	28, 30, 34	mpbir2and 952	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm ℂ))
36	15	simp2d 1036	. . 3 ⊢ (𝜑 → 𝑃 ∈ ∪ 𝐽)
37	11, 36	ffvelcdmd 5783	. 2 ⊢ (𝜑 → (𝐺‘𝑃) ∈ ∪ 𝐾)
38	15	simp3d 1037	. . . . . 6 ⊢ (𝜑 → ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)))
39	38	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)))
40	5	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
41	8	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
42	36	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → 𝑃 ∈ ∪ 𝐽)
43	9	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))
44		simprl 531	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → 𝑢 ∈ 𝐾)
45		simprr 533	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → (𝐺‘𝑃) ∈ 𝑢)
46		icnpimaex 14938	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ ∪ 𝐽) ∧ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢))
47	40, 41, 42, 43, 44, 45, 46	syl33anc 1288	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢))
48		r19.29 2670	. . . . . . 7 ⊢ ((∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)))
49		pm3.45 601	. . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) → ((𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢)))
50	49	imp 124	. . . . . . . 8 ⊢ (((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢))
51	50	reximi 2629	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝐽 ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢))
52	48, 51	syl 14	. . . . . 6 ⊢ ((∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢))
53	11	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝐺:∪ 𝐽⟶∪ 𝐾)
54	53	ffnd 5483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝐺 Fn ∪ 𝐽)
55		simplrl 537	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑣 ∈ 𝐽)
56		elssuni 3921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ⊆ ∪ 𝐽)
57	55, 56	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑣 ⊆ ∪ 𝐽)
58		fnfvima 5889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 Fn ∪ 𝐽 ∧ 𝑣 ⊆ ∪ 𝐽 ∧ (𝐹‘𝑘) ∈ 𝑣) → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣))
59	58	3expia 1231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn ∪ 𝐽 ∧ 𝑣 ⊆ ∪ 𝐽) → ((𝐹‘𝑘) ∈ 𝑣 → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣)))
60	54, 57, 59	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣)))
61	23	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → 𝐹:dom 𝐹⟶∪ 𝐽)
62		fvco3 5717	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:dom 𝐹⟶∪ 𝐽 ∧ 𝑘 ∈ dom 𝐹) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
63	61, 62	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
64	63	eleq1d 2300	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣) ↔ (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣)))
65	60, 64	sylibrd 169	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → ((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣)))
66		simplrr 538	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (𝐺 “ 𝑣) ⊆ 𝑢)
67	66	sseld 3226	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))
68	65, 67	syld 45	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))
69		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑘 ∈ dom 𝐹)
70	26	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → dom (𝐺 ∘ 𝐹) = dom 𝐹)
71	69, 70	eleqtrrd 2311	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑘 ∈ dom (𝐺 ∘ 𝐹))
72	68, 71	jctild 316	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → (𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
73	72	expimpd 363	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → (𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
74	73	ralimdv 2600	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
75	74	reximdv 2633	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
76	75	expr 375	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → ((𝐺 “ 𝑣) ⊆ 𝑢 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))))
77	76	com23 78	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ((𝐺 “ 𝑣) ⊆ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))))
78	77	impd 254	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → ((∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
79	78	rexlimdva 2650	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → (∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
80	52, 79	syl5 32	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ((∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
81	39, 47, 80	mp2and 433	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))
82	81	expr 375	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
83	82	ralrimiva 2605	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐾 ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))
84	8, 12, 13	lmbr2 14941	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm ℂ) ∧ (𝐺‘𝑃) ∈ ∪ 𝐾 ∧ ∀𝑢 ∈ 𝐾 ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))))
85	35, 37, 83, 84	mpbir3and 1206	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200  ∪ cuni 3893   class class class wbr 4088  dom cdm 4725   “ cima 4728   ∘ ccom 4729   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018   ↑_pm cpm 6818  ℂcc 8030  1c1 8033  ℕcn 9143  ℤ_≥cuz 9755  Topctop 14724  TopOnctopon 14737   CnP ccnp 14913  ⇝_𝑡clm 14914

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-pm 6820  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-top 14725  df-topon 14738  df-cnp 14916  df-lm 14917

This theorem is referenced by:  lmcn  14978