Proof of Theorem curry1
| Step | Hyp | Ref
| Expression |
| 1 | | fnfun 6668 |
. . . . 5
⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹) |
| 2 | | 2ndconst 8126 |
. . . . . 6
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V) |
| 3 | | dff1o3 6854 |
. . . . . . 7
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun ◡(2nd ↾ ({𝐶} × V)))) |
| 4 | 3 | simprbi 496 |
. . . . . 6
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
→ Fun ◡(2nd ↾
({𝐶} ×
V))) |
| 5 | 2, 4 | syl 17 |
. . . . 5
⊢ (𝐶 ∈ 𝐴 → Fun ◡(2nd ↾ ({𝐶} × V))) |
| 6 | | funco 6606 |
. . . . 5
⊢ ((Fun
𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))) |
| 7 | 1, 5, 6 | syl2an 596 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))) |
| 8 | | dmco 6274 |
. . . . 5
⊢ dom
(𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) |
| 9 | | fndm 6671 |
. . . . . . . 8
⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) |
| 10 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵)) |
| 11 | 10 | imaeq2d 6078 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))) |
| 12 | | imacnvcnv 6226 |
. . . . . . . . 9
⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) |
| 13 | | df-ima 5698 |
. . . . . . . . 9
⊢
((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) |
| 14 | | resres 6010 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
| 15 | 14 | rneqi 5948 |
. . . . . . . . 9
⊢ ran
((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
| 16 | 12, 13, 15 | 3eqtri 2769 |
. . . . . . . 8
⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
| 17 | | inxp 5842 |
. . . . . . . . . . . . 13
⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) |
| 18 | | incom 4209 |
. . . . . . . . . . . . . . 15
⊢ (V ∩
𝐵) = (𝐵 ∩ V) |
| 19 | | inv1 4398 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∩ V) = 𝐵 |
| 20 | 18, 19 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ (V ∩
𝐵) = 𝐵 |
| 21 | 20 | xpeq2i 5712 |
. . . . . . . . . . . . 13
⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵) |
| 22 | 17, 21 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵) |
| 23 | | snssi 4808 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴) |
| 24 | | dfss2 3969 |
. . . . . . . . . . . . . 14
⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶}) |
| 25 | 23, 24 | sylib 218 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶}) |
| 26 | 25 | xpeq1d 5714 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵)) |
| 27 | 22, 26 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵)) |
| 28 | 27 | reseq2d 5997 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵))) |
| 29 | 28 | rneqd 5949 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵))) |
| 30 | | 2ndconst 8126 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵) |
| 31 | | f1ofo 6855 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾
({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵) |
| 32 | | forn 6823 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵) |
| 33 | 30, 31, 32 | 3syl 18 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵) |
| 34 | 29, 33 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵) |
| 35 | 16, 34 | eqtrid 2789 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐴 → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵) |
| 36 | 35 | adantl 481 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵) |
| 37 | 11, 36 | eqtrd 2777 |
. . . . 5
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵) |
| 38 | 8, 37 | eqtrid 2789 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵) |
| 39 | | curry1.1 |
. . . . . 6
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) |
| 40 | 39 | fneq1i 6665 |
. . . . 5
⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵) |
| 41 | | df-fn 6564 |
. . . . 5
⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵)) |
| 42 | 40, 41 | bitri 275 |
. . . 4
⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵)) |
| 43 | 7, 38, 42 | sylanbrc 583 |
. . 3
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵) |
| 44 | | dffn5 6967 |
. . 3
⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥))) |
| 45 | 43, 44 | sylib 218 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥))) |
| 46 | 39 | fveq1i 6907 |
. . . . 5
⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) |
| 47 | | dff1o4 6856 |
. . . . . . . 8
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V)) |
| 48 | 2, 47 | sylib 218 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V)) |
| 49 | | fvco2 7006 |
. . . . . . . 8
⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
| 50 | 49 | elvd 3486 |
. . . . . . 7
⊢ (◡(2nd ↾ ({𝐶} × V)) Fn V → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
| 51 | 48, 50 | simpl2im 503 |
. . . . . 6
⊢ (𝐶 ∈ 𝐴 → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
| 52 | 51 | ad2antlr 727 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
| 53 | 46, 52 | eqtrid 2789 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
| 54 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V) |
| 55 | | snidg 4660 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶}) |
| 56 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
| 57 | | opelxp 5721 |
. . . . . . . . . . 11
⊢
(〈𝐶, 𝑥〉 ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V)) |
| 58 | 55, 56, 57 | sylanblrc 590 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) |
| 59 | 58 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) |
| 60 | 54, 59 | jca 511 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
∧ 〈𝐶, 𝑥〉 ∈ ({𝐶} × V))) |
| 61 | 58 | fvresd 6926 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = (2nd ‘〈𝐶, 𝑥〉)) |
| 62 | | op2ndg 8027 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd
‘〈𝐶, 𝑥〉) = 𝑥) |
| 63 | 62 | elvd 3486 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ‘〈𝐶, 𝑥〉) = 𝑥) |
| 64 | 61, 63 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥) |
| 65 | 64 | adantr 480 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥) |
| 66 | | f1ocnvfv 7298 |
. . . . . . . 8
⊢
(((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
∧ 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) →
(((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥 → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = 〈𝐶, 𝑥〉)) |
| 67 | 60, 65, 66 | sylc 65 |
. . . . . . 7
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = 〈𝐶, 𝑥〉) |
| 68 | 67 | fveq2d 6910 |
. . . . . 6
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘〈𝐶, 𝑥〉)) |
| 69 | 68 | adantll 714 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘〈𝐶, 𝑥〉)) |
| 70 | | df-ov 7434 |
. . . . 5
⊢ (𝐶𝐹𝑥) = (𝐹‘〈𝐶, 𝑥〉) |
| 71 | 69, 70 | eqtr4di 2795 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥)) |
| 72 | 53, 71 | eqtrd 2777 |
. . 3
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥)) |
| 73 | 72 | mpteq2dva 5242 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
| 74 | 45, 73 | eqtrd 2777 |
1
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |