Theorem cnptopco 14945

Description: The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹‘𝑃) is continuous at 𝑃. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cnptopco	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Proof of Theorem cnptopco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1027	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐾 ∈ Top)
2		toptopon2 14742	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
3	1, 2	sylib 122	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
4		simpl3 1028	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐿 ∈ Top)
5		toptopon2 14742	. . . . 5 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
6	4, 5	sylib 122	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐿 ∈ (TopOn‘∪ 𝐿))
7		simprr 533	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))
8		cnpf2 14930	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐺:∪ 𝐾⟶∪ 𝐿)
9	3, 6, 7, 8	syl3anc 1273	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐺:∪ 𝐾⟶∪ 𝐿)
10		simpl1 1026	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐽 ∈ Top)
11		toptopon2 14742	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
12	10, 11	sylib 122	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐽 ∈ (TopOn‘∪ 𝐽))
13		simprl 531	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
14		cnpf2 14930	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
15	12, 3, 13, 14	syl3anc 1273	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝐹:∪ 𝐽⟶∪ 𝐾)
16		fco 5500	. . 3 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
17	9, 15, 16	syl2anc 411	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
18	3	adantr 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
19	6	adantr 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
20		cnprcl2k 14929	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ ∪ 𝐽)
21	12, 1, 13, 20	syl3anc 1273	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → 𝑃 ∈ ∪ 𝐽)
22	15, 21	ffvelcdmd 5783	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → (𝐹‘𝑃) ∈ ∪ 𝐾)
23	22	adantr 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → (𝐹‘𝑃) ∈ ∪ 𝐾)
24	7	adantr 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))
25		simprl 531	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝑧 ∈ 𝐿)
26		fvco3 5717	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ 𝑃 ∈ ∪ 𝐽) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
27	15, 21, 26	syl2anc 411	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
28	27	adantr 276	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
29		simprr 533	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)
30	28, 29	eqeltrrd 2309	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹‘𝑃)) ∈ 𝑧)
31		icnpimaex 14934	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝐹‘𝑃) ∈ ∪ 𝐾) ∧ (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) ∧ 𝑧 ∈ 𝐿 ∧ (𝐺‘(𝐹‘𝑃)) ∈ 𝑧)) → ∃𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))
32	18, 19, 23, 24, 25, 30, 31	syl33anc 1288	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))
33	12	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝐽 ∈ (TopOn‘∪ 𝐽))
34	3	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
35	21	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝑃 ∈ ∪ 𝐽)
36		simplll 535	. . . . . . . 8 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
37	36	adantlll 480	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
38		simprl 531	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝑦 ∈ 𝐾)
39		simprrl 541	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (𝐹‘𝑃) ∈ 𝑦)
40		icnpimaex 14934	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ ∪ 𝐽) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
41	33, 34, 35, 37, 38, 39, 40	syl33anc 1288	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
42		imaco 5242	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ 𝑥) = (𝐺 “ (𝐹 “ 𝑥))
43		imass2 5112	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹 “ 𝑥)) ⊆ (𝐺 “ 𝑦))
44	42, 43	eqsstrid 3273	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦))
45		simprrr 542	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (𝐺 “ 𝑦) ⊆ 𝑧)
46		sstr2 3234	. . . . . . . . . 10 ⊢ (((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦) → ((𝐺 “ 𝑦) ⊆ 𝑧 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
47	44, 45, 46	syl2imc 39	. . . . . . . . 9 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
48	47	adantlll 480	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
49	48	anim2d 337	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
50	49	reximdv 2633	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
51	41, 50	mpd 13	. . . . 5 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
52	32, 51	rexlimddv 2655	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
53	52	expr 375	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) ∧ 𝑧 ∈ 𝐿) → (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
54	53	ralrimiva 2605	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
55		iscnp 14922	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑃 ∈ ∪ 𝐽) → ((𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃) ↔ ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))))
56	12, 6, 21, 55	syl3anc 1273	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → ((𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃) ↔ ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))))
57	17, 54, 56	mpbir2and 952	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200  ∪ cuni 3893   “ cima 4728   ∘ ccom 4729  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  Topctop 14720  TopOnctopon 14733   CnP ccnp 14909

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

This theorem depends on definitions:  df-bi 117  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-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-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  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-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cnp 14912

This theorem is referenced by: (None)