| Step | Hyp | Ref
| Expression |
| 1 | | elfm 23955 |
. 2
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 2 | | ssfg 23880 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵)) |
| 3 | | elfm2.l |
. . . . . . . . . 10
⊢ 𝐿 = (𝑌filGen𝐵) |
| 4 | 2, 3 | sseqtrrdi 4025 |
. . . . . . . . 9
⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿) |
| 5 | 4 | sselda 3983 |
. . . . . . . 8
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐿) |
| 6 | 5 | adantrr 717 |
. . . . . . 7
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿) |
| 7 | 6 | 3ad2antl2 1187 |
. . . . . 6
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐿) |
| 8 | | simprr 773 |
. . . . . 6
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑦) ⊆ 𝐴) |
| 9 | | imaeq2 6074 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦)) |
| 10 | 9 | sseq1d 4015 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ⊆ 𝐴 ↔ (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 11 | 10 | rspcev 3622 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐿 ∧ (𝐹 “ 𝑦) ⊆ 𝐴) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴) |
| 12 | 7, 8, 11 | syl2anc 584 |
. . . . 5
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑦 ∈ 𝐵 ∧ (𝐹 “ 𝑦) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴) |
| 13 | 12 | rexlimdvaa 3156 |
. . . 4
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)) |
| 14 | 3 | eleq2i 2833 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐿 ↔ 𝑥 ∈ (𝑌filGen𝐵)) |
| 15 | | elfg 23879 |
. . . . . . . 8
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ (𝑌filGen𝐵) ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))) |
| 16 | 14, 15 | bitrid 283 |
. . . . . . 7
⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))) |
| 17 | 16 | 3ad2ant2 1135 |
. . . . . 6
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 ↔ (𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))) |
| 18 | | imass2 6120 |
. . . . . . . . . . 11
⊢ (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥)) |
| 19 | | sstr2 3990 |
. . . . . . . . . . . . 13
⊢ ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝐴 → (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 20 | 19 | com12 32 |
. . . . . . . . . . . 12
⊢ ((𝐹 “ 𝑥) ⊆ 𝐴 → ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 21 | 20 | ad2antll 729 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → ((𝐹 “ 𝑦) ⊆ (𝐹 “ 𝑥) → (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 22 | 18, 21 | syl5 34 |
. . . . . . . . . 10
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → (𝑦 ⊆ 𝑥 → (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 23 | 22 | reximdv 3170 |
. . . . . . . . 9
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 24 | 23 | expr 456 |
. . . . . . . 8
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ⊆ 𝑌) → ((𝐹 “ 𝑥) ⊆ 𝐴 → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 25 | 24 | com23 86 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 26 | 25 | expimpd 453 |
. . . . . 6
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑥 ⊆ 𝑌 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 27 | 17, 26 | sylbid 240 |
. . . . 5
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ 𝐿 → ((𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴))) |
| 28 | 27 | rexlimdv 3153 |
. . . 4
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴)) |
| 29 | 13, 28 | impbid 212 |
. . 3
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴 ↔ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴)) |
| 30 | 29 | anbi2d 630 |
. 2
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))) |
| 31 | 1, 30 | bitrd 279 |
1
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))) |