Step | Hyp | Ref
| Expression |
1 | | cnf2 20863 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) |
2 | 1 | 3expia 1259 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)) |
3 | | elpwi 4117 |
. . . . . . 7
⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) |
4 | 3 | adantl 481 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋) |
5 | | toponuni 20542 |
. . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
6 | 5 | ad2antrr 758 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽) |
7 | 4, 6 | sseqtrd 3604 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ ∪ 𝐽) |
8 | | eqid 2610 |
. . . . . . 7
⊢ ∪ 𝐽 =
∪ 𝐽 |
9 | 8 | cnclsi 20886 |
. . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))) |
10 | 9 | expcom 450 |
. . . . 5
⊢ (𝑥 ⊆ ∪ 𝐽
→ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))) |
11 | 7, 10 | syl 17 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))) |
12 | 11 | ralrimdva 2952 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))) |
13 | 2, 12 | jcad 554 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))) |
14 | | cnvimass 5404 |
. . . . . . . . 9
⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹 |
15 | | fdm 5964 |
. . . . . . . . . 10
⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋) |
16 | 15 | ad2antlr 759 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋) |
17 | 14, 16 | syl5sseq 3616 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋) |
18 | | toponmax 20543 |
. . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
19 | 18 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 ∈ 𝐽) |
20 | | elpw2g 4754 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐽 → ((◡𝐹 “ 𝑦) ∈ 𝒫 𝑋 ↔ (◡𝐹 “ 𝑦) ⊆ 𝑋)) |
21 | 19, 20 | syl 17 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((◡𝐹 “ 𝑦) ∈ 𝒫 𝑋 ↔ (◡𝐹 “ 𝑦) ⊆ 𝑋)) |
22 | 17, 21 | mpbird 246 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ∈ 𝒫 𝑋) |
23 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) |
24 | 23 | imaeq2d 5385 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)))) |
25 | | imaeq2 5381 |
. . . . . . . . . 10
⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦))) |
26 | 25 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐾)‘(𝐹 “ 𝑥)) = ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦)))) |
27 | 24, 26 | sseq12d 3597 |
. . . . . . . 8
⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))))) |
28 | 27 | rspcv 3278 |
. . . . . . 7
⊢ ((◡𝐹 “ 𝑦) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))))) |
29 | 22, 28 | syl 17 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))))) |
30 | | topontop 20541 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) |
31 | 30 | ad3antlr 763 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top) |
32 | | elpwi 4117 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝒫 𝑌 → 𝑦 ⊆ 𝑌) |
33 | 32 | adantl 481 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ 𝑌) |
34 | | toponuni 20542 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) |
35 | 34 | ad3antlr 763 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾) |
36 | 33, 35 | sseqtrd 3604 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ ∪ 𝐾) |
37 | | ffun 5961 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹) |
38 | 37 | ad2antlr 759 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹) |
39 | | funimacnv 5884 |
. . . . . . . . . . 11
⊢ (Fun
𝐹 → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹)) |
40 | 38, 39 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹)) |
41 | | inss1 3795 |
. . . . . . . . . 10
⊢ (𝑦 ∩ ran 𝐹) ⊆ 𝑦 |
42 | 40, 41 | syl6eqss 3618 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦) |
43 | | eqid 2610 |
. . . . . . . . . 10
⊢ ∪ 𝐾 =
∪ 𝐾 |
44 | 43 | clsss 20668 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾
∧ (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦)) |
45 | 31, 36, 42, 44 | syl3anc 1318 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦)) |
46 | | sstr2 3575 |
. . . . . . . 8
⊢ ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → (((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))) |
47 | 45, 46 | syl5com 31 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))) |
48 | | topontop 20541 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
49 | 48 | ad3antrrr 762 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top) |
50 | 5 | ad3antrrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = ∪ 𝐽) |
51 | 16, 50 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = ∪ 𝐽) |
52 | 14, 51 | syl5sseq 3616 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽) |
53 | 8 | clsss3 20673 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽) |
54 | 49, 52, 53 | syl2anc 691 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽) |
55 | 54, 51 | sseqtr4d 3605 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ dom 𝐹) |
56 | | funimass3 6241 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))) |
57 | 38, 55, 56 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))) |
58 | 47, 57 | sylibd 228 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))) |
59 | 29, 58 | syld 46 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))) |
60 | 59 | ralrimdva 2952 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))) |
61 | 60 | imdistanda 725 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))) |
62 | | cncls2 20887 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))) |
63 | 61, 62 | sylibrd 248 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))) |
64 | 13, 63 | impbid 201 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))) |