Step | Hyp | Ref
| Expression |
1 | | foima 6120 |
. . . 4
⊢ (𝐹:𝑌–onto→𝑋 → (𝐹 “ 𝑌) = 𝑋) |
2 | 1 | adantl 482 |
. . 3
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) = 𝑋) |
3 | | fofun 6116 |
. . . 4
⊢ (𝐹:𝑌–onto→𝑋 → Fun 𝐹) |
4 | | elfvdm 6220 |
. . . 4
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas) |
5 | | funimaexg 5975 |
. . . 4
⊢ ((Fun
𝐹 ∧ 𝑌 ∈ dom fBas) → (𝐹 “ 𝑌) ∈ V) |
6 | 3, 4, 5 | syl2anr 495 |
. . 3
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) ∈ V) |
7 | 2, 6 | eqeltrrd 2702 |
. 2
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ∈ V) |
8 | | fof 6115 |
. . . . 5
⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋) |
9 | | elfm2.l |
. . . . . 6
⊢ 𝐿 = (𝑌filGen𝐵) |
10 | 9 | elfm2 21752 |
. . . . 5
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))) |
11 | 8, 10 | syl3an3 1361 |
. . . 4
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))) |
12 | | fgcl 21682 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌)) |
13 | 9, 12 | syl5eqel 2705 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌)) |
14 | 13 | 3ad2ant2 1083 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝐿 ∈ (Fil‘𝑌)) |
15 | 14 | ad2antrr 762 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌)) |
16 | | simprl 794 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿) |
17 | | cnvimass 5485 |
. . . . . . . . . . . 12
⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 |
18 | | fofn 6117 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝑌–onto→𝑋 → 𝐹 Fn 𝑌) |
19 | | fndm 5990 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn 𝑌 → dom 𝐹 = 𝑌) |
20 | 18, 19 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑌–onto→𝑋 → dom 𝐹 = 𝑌) |
21 | 17, 20 | syl5sseq 3653 |
. . . . . . . . . . 11
⊢ (𝐹:𝑌–onto→𝑋 → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
22 | 21 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
23 | 22 | ad2antrr 762 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
24 | 3 | 3ad2ant3 1084 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → Fun 𝐹) |
25 | 24 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → Fun 𝐹) |
26 | 9 | eleq2i 2693 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝐿 ↔ 𝑦 ∈ (𝑌filGen𝐵)) |
27 | | elfg 21675 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
28 | 27 | 3ad2ant2 1083 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
29 | 28 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
30 | 26, 29 | syl5bb 272 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ 𝐿 ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
31 | 30 | simprbda 653 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ 𝑌) |
32 | | sseq2 3627 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹 ↔ 𝑦 ⊆ 𝑌)) |
33 | 32 | biimpar 502 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
𝐹 = 𝑌 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
34 | 20, 33 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
35 | 34 | 3ad2antl3 1225 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
36 | 35 | adantlr 751 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
37 | 31, 36 | syldan 487 |
. . . . . . . . . . . 12
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ dom 𝐹) |
38 | | funimass3 6333 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ 𝑦 ⊆ dom 𝐹) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
39 | 25, 37, 38 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
40 | 39 | biimpd 219 |
. . . . . . . . . 10
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 → 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
41 | 40 | impr 649 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (◡𝐹 “ 𝐴)) |
42 | | filss 21657 |
. . . . . . . . 9
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦 ∈ 𝐿 ∧ (◡𝐹 “ 𝐴) ⊆ 𝑌 ∧ 𝑦 ⊆ (◡𝐹 “ 𝐴))) → (◡𝐹 “ 𝐴) ∈ 𝐿) |
43 | 15, 16, 23, 41, 42 | syl13anc 1328 |
. . . . . . . 8
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ∈ 𝐿) |
44 | | foimacnv 6154 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴) |
45 | 44 | eqcomd 2628 |
. . . . . . . . . 10
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
46 | 45 | 3ad2antl3 1225 |
. . . . . . . . 9
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
47 | 46 | adantr 481 |
. . . . . . . 8
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
48 | | imaeq2 5462 |
. . . . . . . . . 10
⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝐴))) |
49 | 48 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐴 = (𝐹 “ 𝑥) ↔ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴)))) |
50 | 49 | rspcev 3309 |
. . . . . . . 8
⊢ (((◡𝐹 “ 𝐴) ∈ 𝐿 ∧ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)) |
51 | 43, 47, 50 | syl2anc 693 |
. . . . . . 7
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)) |
52 | 51 | rexlimdvaa 3032 |
. . . . . 6
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
53 | 52 | expimpd 629 |
. . . . 5
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
54 | | simprr 796 |
. . . . . . . 8
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 = (𝐹 “ 𝑥)) |
55 | | imassrn 5477 |
. . . . . . . . 9
⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹 |
56 | | forn 6118 |
. . . . . . . . . . 11
⊢ (𝐹:𝑌–onto→𝑋 → ran 𝐹 = 𝑋) |
57 | 56 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ran 𝐹 = 𝑋) |
58 | 57 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ran 𝐹 = 𝑋) |
59 | 55, 58 | syl5sseq 3653 |
. . . . . . . 8
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐹 “ 𝑥) ⊆ 𝑋) |
60 | 54, 59 | eqsstrd 3639 |
. . . . . . 7
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 ⊆ 𝑋) |
61 | | eqimss2 3658 |
. . . . . . . . 9
⊢ (𝐴 = (𝐹 “ 𝑥) → (𝐹 “ 𝑥) ⊆ 𝐴) |
62 | | imaeq2 5462 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (𝐹 “ 𝑦) = (𝐹 “ 𝑥)) |
63 | 62 | sseq1d 3632 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴)) |
64 | 63 | rspcev 3309 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐿 ∧ (𝐹 “ 𝑥) ⊆ 𝐴) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
65 | 61, 64 | sylan2 491 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
66 | 65 | adantl 482 |
. . . . . . 7
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
67 | 60, 66 | jca 554 |
. . . . . 6
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)) |
68 | 67 | rexlimdvaa 3032 |
. . . . 5
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))) |
69 | 53, 68 | impbid 202 |
. . . 4
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
70 | 11, 69 | bitrd 268 |
. . 3
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
71 | 70 | 3coml 1272 |
. 2
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋 ∧ 𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
72 | 7, 71 | mpd3an3 1425 |
1
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |