Proof of Theorem uhgrimisgrgriclem
Step | Hyp | Ref
| Expression |
1 | | fveq2 6920 |
. . . . . 6
⊢ (𝑘 = (◡𝐼‘𝐽) → (𝐺‘𝑘) = (𝐺‘(◡𝐼‘𝐽))) |
2 | 1 | sseq1d 4040 |
. . . . 5
⊢ (𝑘 = (◡𝐼‘𝐽) → ((𝐺‘𝑘) ⊆ 𝑁 ↔ (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
3 | | fveqeq2 6929 |
. . . . 5
⊢ (𝑘 = (◡𝐼‘𝐽) → ((𝐼‘𝑘) = 𝐽 ↔ (𝐼‘(◡𝐼‘𝐽)) = 𝐽)) |
4 | 2, 3 | anbi12d 631 |
. . . 4
⊢ (𝑘 = (◡𝐼‘𝐽) → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) ↔ ((𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁 ∧ (𝐼‘(◡𝐼‘𝐽)) = 𝐽))) |
5 | | simpr 484 |
. . . . . 6
⊢ ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → 𝐼:𝐴–1-1-onto→𝐵) |
6 | 5 | 3ad2ant2 1134 |
. . . . 5
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → 𝐼:𝐴–1-1-onto→𝐵) |
7 | | simpl 482 |
. . . . 5
⊢ ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → 𝐽 ∈ 𝐵) |
8 | | f1ocnvdm 7321 |
. . . . 5
⊢ ((𝐼:𝐴–1-1-onto→𝐵 ∧ 𝐽 ∈ 𝐵) → (◡𝐼‘𝐽) ∈ 𝐴) |
9 | 6, 7, 8 | syl2an 595 |
. . . 4
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → (◡𝐼‘𝐽) ∈ 𝐴) |
10 | | 2fveq3 6925 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (◡𝐼‘𝐽) → (𝐻‘(𝐼‘𝑖)) = (𝐻‘(𝐼‘(◡𝐼‘𝐽)))) |
11 | | fveq2 6920 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (◡𝐼‘𝐽) → (𝐺‘𝑖) = (𝐺‘(◡𝐼‘𝐽))) |
12 | 11 | imaeq2d 6089 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (◡𝐼‘𝐽) → (𝐹 “ (𝐺‘𝑖)) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽)))) |
13 | 10, 12 | eqeq12d 2756 |
. . . . . . . . . . . 12
⊢ (𝑖 = (◡𝐼‘𝐽) → ((𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) ↔ (𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))))) |
14 | 13 | rspcv 3631 |
. . . . . . . . . . 11
⊢ ((◡𝐼‘𝐽) ∈ 𝐴 → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))))) |
15 | 14 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))))) |
16 | 7 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → 𝐽 ∈ 𝐵) |
17 | | f1ocnvfv2 7313 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼:𝐴–1-1-onto→𝐵 ∧ 𝐽 ∈ 𝐵) → (𝐼‘(◡𝐼‘𝐽)) = 𝐽) |
18 | 5, 16, 17 | syl2anr 596 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (𝐼‘(◡𝐼‘𝐽)) = 𝐽) |
19 | 18 | fveqeq2d 6928 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ↔ (𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))))) |
20 | | sseq1 4034 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → ((𝐻‘𝐽) ⊆ (𝐹 “ 𝑁) ↔ (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁))) |
21 | 20 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽)))) → ((𝐻‘𝐽) ⊆ (𝐹 “ 𝑁) ↔ (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁))) |
22 | | f1of1 6861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊) |
23 | 22 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) → 𝐹:𝑉–1-1→𝑊) |
24 | 23 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → 𝐹:𝑉–1-1→𝑊) |
25 | 24 | 3ad2ant1 1133 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → 𝐹:𝑉–1-1→𝑊) |
26 | | simp1lr 1237 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → 𝐺:𝐴⟶𝒫 𝑉) |
27 | | simp1r 1198 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (◡𝐼‘𝐽) ∈ 𝐴) |
28 | 26, 27 | ffvelcdmd 7119 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (𝐺‘(◡𝐼‘𝐽)) ∈ 𝒫 𝑉) |
29 | 28 | elpwid 4631 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑉) |
30 | | simpl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → 𝑁 ⊆ 𝑉) |
31 | 30 | 3ad2ant3 1135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → 𝑁 ⊆ 𝑉) |
32 | | f1imass 7301 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹:𝑉–1-1→𝑊 ∧ ((𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑉 ∧ 𝑁 ⊆ 𝑉)) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) ↔ (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
33 | 25, 29, 31, 32 | syl12anc 836 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) ↔ (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
34 | 33 | biimpd 229 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ 𝐽 ∈ 𝐵 ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
35 | 34 | 3exp 1119 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → (𝐽 ∈ 𝐵 → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
36 | 35 | com24 95 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐽 ∈ 𝐵 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
37 | 36 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽)))) → ((𝐹 “ (𝐺‘(◡𝐼‘𝐽))) ⊆ (𝐹 “ 𝑁) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐽 ∈ 𝐵 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
38 | 21, 37 | sylbid 240 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽)))) → ((𝐻‘𝐽) ⊆ (𝐹 “ 𝑁) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐽 ∈ 𝐵 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
39 | 38 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → ((𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → ((𝐻‘𝐽) ⊆ (𝐹 “ 𝑁) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐽 ∈ 𝐵 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))))) |
40 | 39 | com25 99 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → (𝐽 ∈ 𝐵 → ((𝐻‘𝐽) ⊆ (𝐹 “ 𝑁) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → ((𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))))) |
41 | 40 | imp42 426 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → ((𝐻‘𝐽) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
42 | 19, 41 | sylbid 240 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
43 | 42 | ex 412 |
. . . . . . . . . . . . 13
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))) |
44 | 43 | com23 86 |
. . . . . . . . . . . 12
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))) |
45 | 44 | ex 412 |
. . . . . . . . . . 11
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
46 | 45 | com23 86 |
. . . . . . . . . 10
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → ((𝐻‘(𝐼‘(◡𝐼‘𝐽))) = (𝐹 “ (𝐺‘(◡𝐼‘𝐽))) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
47 | 15, 46 | syld 47 |
. . . . . . . . 9
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (◡𝐼‘𝐽) ∈ 𝐴) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)))) |
48 | 47 | ex 412 |
. . . . . . . 8
⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) → ((◡𝐼‘𝐽) ∈ 𝐴 → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))))) |
49 | 48 | com25 99 |
. . . . . . 7
⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) → ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ((◡𝐼‘𝐽) ∈ 𝐴 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁))))) |
50 | 49 | 3imp1 1347 |
. . . . . 6
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → ((◡𝐼‘𝐽) ∈ 𝐴 → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁)) |
51 | 9, 50 | mpd 15 |
. . . . 5
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → (𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁) |
52 | 6, 7, 17 | syl2an 595 |
. . . . 5
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → (𝐼‘(◡𝐼‘𝐽)) = 𝐽) |
53 | 51, 52 | jca 511 |
. . . 4
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → ((𝐺‘(◡𝐼‘𝐽)) ⊆ 𝑁 ∧ (𝐼‘(◡𝐼‘𝐽)) = 𝐽)) |
54 | 4, 9, 53 | rspcedvdw 3638 |
. . 3
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) → ∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) |
55 | 54 | ex 412 |
. 2
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) → ∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽))) |
56 | | f1of 6862 |
. . . . . . . 8
⊢ (𝐼:𝐴–1-1-onto→𝐵 → 𝐼:𝐴⟶𝐵) |
57 | 56 | adantl 481 |
. . . . . . 7
⊢ ((𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) → 𝐼:𝐴⟶𝐵) |
58 | 57 | 3ad2ant2 1134 |
. . . . . 6
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → 𝐼:𝐴⟶𝐵) |
59 | 58 | 3ad2ant1 1133 |
. . . . 5
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → 𝐼:𝐴⟶𝐵) |
60 | | simp2 1137 |
. . . . 5
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → 𝑘 ∈ 𝐴) |
61 | 59, 60 | ffvelcdmd 7119 |
. . . 4
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → (𝐼‘𝑘) ∈ 𝐵) |
62 | | 2fveq3 6925 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑘 → (𝐻‘(𝐼‘𝑖)) = (𝐻‘(𝐼‘𝑘))) |
63 | | fveq2 6920 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑘 → (𝐺‘𝑖) = (𝐺‘𝑘)) |
64 | 63 | imaeq2d 6089 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑘 → (𝐹 “ (𝐺‘𝑖)) = (𝐹 “ (𝐺‘𝑘))) |
65 | 62, 64 | eqeq12d 2756 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑘 → ((𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) ↔ (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘)))) |
66 | 65 | rspcv 3631 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝐴 → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘)))) |
67 | 66 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘)))) |
68 | | simp3 1138 |
. . . . . . . . . . . 12
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) ∧ (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘))) → (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘))) |
69 | | imass2 6132 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘𝑘) ⊆ 𝑁 → (𝐹 “ (𝐺‘𝑘)) ⊆ (𝐹 “ 𝑁)) |
70 | 69 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐹 “ (𝐺‘𝑘)) ⊆ (𝐹 “ 𝑁)) |
71 | 70 | 3ad2ant2 1134 |
. . . . . . . . . . . 12
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) ∧ (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘))) → (𝐹 “ (𝐺‘𝑘)) ⊆ (𝐹 “ 𝑁)) |
72 | 68, 71 | eqsstrd 4047 |
. . . . . . . . . . 11
⊢
(((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) ∧ (𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘))) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)) |
73 | 72 | 3exp 1119 |
. . . . . . . . . 10
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → ((𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘)) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)))) |
74 | 73 | com23 86 |
. . . . . . . . 9
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) → ((𝐻‘(𝐼‘𝑘)) = (𝐹 “ (𝐺‘𝑘)) → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)))) |
75 | 67, 74 | syld 47 |
. . . . . . . 8
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐴) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)))) |
76 | 75 | ex 412 |
. . . . . . 7
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (𝑘 ∈ 𝐴 → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁))))) |
77 | 76 | com23 86 |
. . . . . 6
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵)) → (∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖)) → (𝑘 ∈ 𝐴 → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁))))) |
78 | 77 | 3impia 1117 |
. . . . 5
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → (𝑘 ∈ 𝐴 → (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)))) |
79 | 78 | 3imp 1111 |
. . . 4
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)) |
80 | | eleq1 2832 |
. . . . . . 7
⊢ ((𝐼‘𝑘) = 𝐽 → ((𝐼‘𝑘) ∈ 𝐵 ↔ 𝐽 ∈ 𝐵)) |
81 | | fveq2 6920 |
. . . . . . . 8
⊢ ((𝐼‘𝑘) = 𝐽 → (𝐻‘(𝐼‘𝑘)) = (𝐻‘𝐽)) |
82 | 81 | sseq1d 4040 |
. . . . . . 7
⊢ ((𝐼‘𝑘) = 𝐽 → ((𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁) ↔ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) |
83 | 80, 82 | anbi12d 631 |
. . . . . 6
⊢ ((𝐼‘𝑘) = 𝐽 → (((𝐼‘𝑘) ∈ 𝐵 ∧ (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)) ↔ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)))) |
84 | 83 | adantl 481 |
. . . . 5
⊢ (((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (((𝐼‘𝑘) ∈ 𝐵 ∧ (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)) ↔ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)))) |
85 | 84 | 3ad2ant3 1135 |
. . . 4
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → (((𝐼‘𝑘) ∈ 𝐵 ∧ (𝐻‘(𝐼‘𝑘)) ⊆ (𝐹 “ 𝑁)) ↔ (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)))) |
86 | 61, 79, 85 | mpbi2and 711 |
. . 3
⊢ ((((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) ∧ 𝑘 ∈ 𝐴 ∧ ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)) → (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁))) |
87 | 86 | rexlimdv3a 3165 |
. 2
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → (∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽) → (𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)))) |
88 | 55, 87 | impbid 212 |
1
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) ↔ ∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽))) |