Proof of Theorem subsaliuncllem
| Step | Hyp | Ref
| Expression |
| 1 | | subsaliuncllem.e |
. . 3
⊢ 𝐸 = (𝐻 ∘ 𝐺) |
| 2 | | subsaliuncllem.h |
. . . . . . 7
⊢ (𝜑 → 𝐻 Fn ran 𝐺) |
| 3 | | subsaliuncllem.f |
. . . . . . . 8
⊢
Ⅎ𝑦𝜑 |
| 4 | | vex 3484 |
. . . . . . . . . . . . . 14
⊢ 𝑦 ∈ V |
| 5 | | subsaliuncllem.g |
. . . . . . . . . . . . . . 15
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 6 | 5 | elrnmpt 5969 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) |
| 7 | 4, 6 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 8 | 7 | biimpi 216 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 9 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 10 | | ssrab2 4080 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ⊆ 𝑆 |
| 11 | 10 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ⊆ 𝑆) |
| 12 | 9, 11 | eqsstrd 4018 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆) |
| 13 | 12 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆)) |
| 14 | 13 | rexlimiv 3148 |
. . . . . . . . . . . . 13
⊢
(∃𝑛 ∈
ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆) |
| 15 | 14 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ran 𝐺 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆)) |
| 16 | 8, 15 | mpd 15 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ⊆ 𝑆) |
| 17 | 16 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ⊆ 𝑆) |
| 18 | | subsaliuncllem.y |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) |
| 19 | 18 | r19.21bi 3251 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑦) |
| 20 | 17, 19 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑆) |
| 21 | 20 | ex 412 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻‘𝑦) ∈ 𝑆)) |
| 22 | 3, 21 | ralrimi 3257 |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆) |
| 23 | 2, 22 | jca 511 |
. . . . . 6
⊢ (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆)) |
| 24 | | ffnfv 7139 |
. . . . . 6
⊢ (𝐻:ran 𝐺⟶𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆)) |
| 25 | 23, 24 | sylibr 234 |
. . . . 5
⊢ (𝜑 → 𝐻:ran 𝐺⟶𝑆) |
| 26 | | eqid 2737 |
. . . . . . . . 9
⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} |
| 27 | | subsaliuncllem.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ 𝑉) |
| 28 | 26, 27 | rabexd 5340 |
. . . . . . . 8
⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V) |
| 29 | 28 | ralrimivw 3150 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V) |
| 30 | 5 | fnmpt 6708 |
. . . . . . 7
⊢
(∀𝑛 ∈
ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → 𝐺 Fn ℕ) |
| 31 | 29, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺 Fn ℕ) |
| 32 | | dffn3 6748 |
. . . . . 6
⊢ (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺) |
| 33 | 31, 32 | sylib 218 |
. . . . 5
⊢ (𝜑 → 𝐺:ℕ⟶ran 𝐺) |
| 34 | | fco 6760 |
. . . . 5
⊢ ((𝐻:ran 𝐺⟶𝑆 ∧ 𝐺:ℕ⟶ran 𝐺) → (𝐻 ∘ 𝐺):ℕ⟶𝑆) |
| 35 | 25, 33, 34 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝐻 ∘ 𝐺):ℕ⟶𝑆) |
| 36 | | nnex 12272 |
. . . . . 6
⊢ ℕ
∈ V |
| 37 | 36 | a1i 11 |
. . . . 5
⊢ (𝜑 → ℕ ∈
V) |
| 38 | 27, 37 | elmapd 8880 |
. . . 4
⊢ (𝜑 → ((𝐻 ∘ 𝐺) ∈ (𝑆 ↑m ℕ) ↔ (𝐻 ∘ 𝐺):ℕ⟶𝑆)) |
| 39 | 35, 38 | mpbird 257 |
. . 3
⊢ (𝜑 → (𝐻 ∘ 𝐺) ∈ (𝑆 ↑m
ℕ)) |
| 40 | 1, 39 | eqeltrid 2845 |
. 2
⊢ (𝜑 → 𝐸 ∈ (𝑆 ↑m
ℕ)) |
| 41 | 33 | ffvelcdmda 7104 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ran 𝐺) |
| 42 | 18 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) |
| 43 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑦 = (𝐺‘𝑛) → (𝐻‘𝑦) = (𝐻‘(𝐺‘𝑛))) |
| 44 | | id 22 |
. . . . . . . . 9
⊢ (𝑦 = (𝐺‘𝑛) → 𝑦 = (𝐺‘𝑛)) |
| 45 | 43, 44 | eleq12d 2835 |
. . . . . . . 8
⊢ (𝑦 = (𝐺‘𝑛) → ((𝐻‘𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))) |
| 46 | 45 | rspcva 3620 |
. . . . . . 7
⊢ (((𝐺‘𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛)) |
| 47 | 41, 42, 46 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛)) |
| 48 | 33 | ffund 6740 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝐺) |
| 49 | 48 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun 𝐺) |
| 50 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
| 51 | 5 | dmeqi 5915 |
. . . . . . . . . . . . 13
⊢ dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) |
| 53 | | dmmptg 6262 |
. . . . . . . . . . . . 13
⊢
(∀𝑛 ∈
ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ℕ) |
| 54 | 29, 53 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ℕ) |
| 55 | 52, 54 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝐺 = ℕ) |
| 56 | 55 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → ℕ = dom 𝐺) |
| 57 | 56 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom 𝐺) |
| 58 | 50, 57 | eleqtrd 2843 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺) |
| 59 | 49, 58, 1 | fvcod 45232 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) = (𝐻‘(𝐺‘𝑛))) |
| 60 | 5 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) |
| 61 | 28 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V) |
| 62 | 60, 61 | fvmpt2d 7029 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 63 | 62 | eqcomd 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = (𝐺‘𝑛)) |
| 64 | 59, 63 | eleq12d 2835 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ↔ (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))) |
| 65 | 47, 64 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) |
| 66 | | ineq1 4213 |
. . . . . . 7
⊢ (𝑥 = (𝐸‘𝑛) → (𝑥 ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷)) |
| 67 | 66 | eqeq2d 2748 |
. . . . . 6
⊢ (𝑥 = (𝐸‘𝑛) → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))) |
| 68 | 67 | elrab 3692 |
. . . . 5
⊢ ((𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ↔ ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))) |
| 69 | 65, 68 | sylib 218 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))) |
| 70 | 69 | simprd 495 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)) |
| 71 | 70 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)) |
| 72 | | fveq1 6905 |
. . . . . 6
⊢ (𝑒 = 𝐸 → (𝑒‘𝑛) = (𝐸‘𝑛)) |
| 73 | 72 | ineq1d 4219 |
. . . . 5
⊢ (𝑒 = 𝐸 → ((𝑒‘𝑛) ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷)) |
| 74 | 73 | eqeq2d 2748 |
. . . 4
⊢ (𝑒 = 𝐸 → ((𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ↔ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))) |
| 75 | 74 | ralbidv 3178 |
. . 3
⊢ (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))) |
| 76 | 75 | rspcev 3622 |
. 2
⊢ ((𝐸 ∈ (𝑆 ↑m ℕ) ∧
∀𝑛 ∈ ℕ
(𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) |
| 77 | 40, 71, 76 | syl2anc 584 |
1
⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) |