| Step | Hyp | Ref
| Expression |
| 1 | | foima 6825 |
. . . 4
⊢ (𝐹:𝑌–onto→𝑋 → (𝐹 “ 𝑌) = 𝑋) |
| 2 | 1 | adantl 481 |
. . 3
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) = 𝑋) |
| 3 | | fofun 6821 |
. . . 4
⊢ (𝐹:𝑌–onto→𝑋 → Fun 𝐹) |
| 4 | | elfvdm 6943 |
. . . 4
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas) |
| 5 | | funimaexg 6653 |
. . . 4
⊢ ((Fun
𝐹 ∧ 𝑌 ∈ dom fBas) → (𝐹 “ 𝑌) ∈ V) |
| 6 | 3, 4, 5 | syl2anr 597 |
. . 3
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐹 “ 𝑌) ∈ V) |
| 7 | 2, 6 | eqeltrrd 2842 |
. 2
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ∈ V) |
| 8 | | fof 6820 |
. . . . 5
⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋) |
| 9 | | elfm2.l |
. . . . . 6
⊢ 𝐿 = (𝑌filGen𝐵) |
| 10 | 9 | elfm2 23956 |
. . . . 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 1166 |
. . . 4
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 12 | | fgcl 23886 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌)) |
| 13 | 9, 12 | eqeltrid 2845 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌)) |
| 14 | 13 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → 𝐿 ∈ (Fil‘𝑌)) |
| 15 | 14 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌)) |
| 16 | | simprl 771 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿) |
| 17 | | cnvimass 6100 |
. . . . . . . . . . . 12
⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 |
| 18 | | fofn 6822 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝑌–onto→𝑋 → 𝐹 Fn 𝑌) |
| 19 | 18 | fndmd 6673 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑌–onto→𝑋 → dom 𝐹 = 𝑌) |
| 20 | 17, 19 | sseqtrid 4026 |
. . . . . . . . . . 11
⊢ (𝐹:𝑌–onto→𝑋 → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
| 21 | 20 | 3ad2ant3 1136 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
| 22 | 21 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ⊆ 𝑌) |
| 23 | 3 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → Fun 𝐹) |
| 24 | 23 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → Fun 𝐹) |
| 25 | 9 | eleq2i 2833 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝐿 ↔ 𝑦 ∈ (𝑌filGen𝐵)) |
| 26 | | elfg 23879 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
| 27 | 26 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
| 28 | 27 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
| 29 | 25, 28 | bitrid 283 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑦 ∈ 𝐿 ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦))) |
| 30 | 29 | simprbda 498 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ 𝑌) |
| 31 | | sseq2 4010 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹 ↔ 𝑦 ⊆ 𝑌)) |
| 32 | 31 | biimpar 477 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
𝐹 = 𝑌 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
| 33 | 19, 32 | sylan 580 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
| 34 | 33 | 3ad2antl3 1188 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
| 35 | 34 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → 𝑦 ⊆ dom 𝐹) |
| 36 | 30, 35 | syldan 591 |
. . . . . . . . . . . 12
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → 𝑦 ⊆ dom 𝐹) |
| 37 | | funimass3 7074 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ 𝑦 ⊆ dom 𝐹) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
| 38 | 24, 36, 37 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
| 39 | 38 | biimpd 229 |
. . . . . . . . . 10
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐿) → ((𝐹 “ 𝑦) ⊆ 𝐴 → 𝑦 ⊆ (◡𝐹 “ 𝐴))) |
| 40 | 39 | impr 454 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (◡𝐹 “ 𝐴)) |
| 41 | | filss 23861 |
. . . . . . . . 9
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦 ∈ 𝐿 ∧ (◡𝐹 “ 𝐴) ⊆ 𝑌 ∧ 𝑦 ⊆ (◡𝐹 “ 𝐴))) → (◡𝐹 “ 𝐴) ∈ 𝐿) |
| 42 | 15, 16, 22, 40, 41 | syl13anc 1374 |
. . . . . . . 8
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (◡𝐹 “ 𝐴) ∈ 𝐿) |
| 43 | | foimacnv 6865 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴) |
| 44 | 43 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝐹:𝑌–onto→𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
| 45 | 44 | 3ad2antl3 1188 |
. . . . . . . . 9
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
| 46 | 45 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) |
| 47 | | imaeq2 6074 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝐹 “ 𝐴) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝐴))) |
| 48 | 47 | rspceeqv 3645 |
. . . . . . . 8
⊢ (((◡𝐹 “ 𝐴) ∈ 𝐿 ∧ 𝐴 = (𝐹 “ (◡𝐹 “ 𝐴))) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)) |
| 49 | 42, 46, 48 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥)) |
| 50 | 49 | rexlimdvaa 3156 |
. . . . . 6
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ 𝐴 ⊆ 𝑋) → (∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
| 51 | 50 | expimpd 453 |
. . . . 5
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
| 52 | | simprr 773 |
. . . . . . . 8
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 = (𝐹 “ 𝑥)) |
| 53 | | imassrn 6089 |
. . . . . . . . 9
⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹 |
| 54 | | forn 6823 |
. . . . . . . . . . 11
⊢ (𝐹:𝑌–onto→𝑋 → ran 𝐹 = 𝑋) |
| 55 | 54 | 3ad2ant3 1136 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ran 𝐹 = 𝑋) |
| 56 | 55 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ran 𝐹 = 𝑋) |
| 57 | 53, 56 | sseqtrid 4026 |
. . . . . . . 8
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐹 “ 𝑥) ⊆ 𝑋) |
| 58 | 52, 57 | eqsstrd 4018 |
. . . . . . 7
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → 𝐴 ⊆ 𝑋) |
| 59 | | eqimss2 4043 |
. . . . . . . . 9
⊢ (𝐴 = (𝐹 “ 𝑥) → (𝐹 “ 𝑥) ⊆ 𝐴) |
| 60 | | imaeq2 6074 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (𝐹 “ 𝑦) = (𝐹 “ 𝑥)) |
| 61 | 60 | sseq1d 4015 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝐹 “ 𝑦) ⊆ 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴)) |
| 62 | 61 | rspcev 3622 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐿 ∧ (𝐹 “ 𝑥) ⊆ 𝐴) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
| 63 | 59, 62 | sylan2 593 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
| 64 | 63 | adantl 481 |
. . . . . . 7
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) |
| 65 | 58, 64 | jca 511 |
. . . . . 6
⊢ (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) ∧ (𝑥 ∈ 𝐿 ∧ 𝐴 = (𝐹 “ 𝑥))) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 66 | 65 | rexlimdvaa 3156 |
. . . . 5
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥) → (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 67 | 51, 66 | impbid 212 |
. . . 4
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐿 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
| 68 | 11, 67 | bitrd 279 |
. . 3
⊢ ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
| 69 | 68 | 3coml 1128 |
. 2
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋 ∧ 𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |
| 70 | 7, 69 | mpd3an3 1464 |
1
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) |