Step | Hyp | Ref
| Expression |
1 | | relco 6108 |
. 2
⊢ Rel
(𝐺 ∘ 𝐹) |
2 | | mptrel 5826 |
. 2
⊢ Rel
(𝑥 ∈ 𝐴 ↦ 𝑇) |
3 | | fmptcof2.3 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝜑 |
4 | | fmptcof2.1 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝐴 |
5 | | fmptcof2.2 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝐵 |
6 | | fmptcof2.4 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) |
7 | 6 | r19.21bi 3249 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵) |
8 | | eqid 2733 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅) |
9 | 3, 4, 5, 7, 8 | fmptdF 31881 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵) |
10 | | fmptcof2.5 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) |
11 | 10 | feq1d 6703 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵)) |
12 | 9, 11 | mpbird 257 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
13 | 12 | ffund 6722 |
. . . . . . . . . 10
⊢ (𝜑 → Fun 𝐹) |
14 | | funbrfv 6943 |
. . . . . . . . . . 11
⊢ (Fun
𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢)) |
15 | 14 | imp 408 |
. . . . . . . . . 10
⊢ ((Fun
𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢) |
16 | 13, 15 | sylan 581 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢) |
17 | 16 | eqcomd 2739 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → 𝑢 = (𝐹‘𝑧)) |
18 | 17 | a1d 25 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝑢𝐺𝑤 → 𝑢 = (𝐹‘𝑧))) |
19 | 18 | expimpd 455 |
. . . . . 6
⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) → 𝑢 = (𝐹‘𝑧))) |
20 | 19 | pm4.71rd 564 |
. . . . 5
⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)))) |
21 | 20 | exbidv 1925 |
. . . 4
⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)))) |
22 | | fvex 6905 |
. . . . . 6
⊢ (𝐹‘𝑧) ∈ V |
23 | | breq2 5153 |
. . . . . . 7
⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧))) |
24 | | breq1 5152 |
. . . . . . 7
⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤)) |
25 | 23, 24 | anbi12d 632 |
. . . . . 6
⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤))) |
26 | 22, 25 | ceqsexv 3526 |
. . . . 5
⊢
(∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)) |
27 | | funfvbrb 7053 |
. . . . . . . . 9
⊢ (Fun
𝐹 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧))) |
28 | 13, 27 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧))) |
29 | 12 | fdmd 6729 |
. . . . . . . . 9
⊢ (𝜑 → dom 𝐹 = 𝐴) |
30 | 29 | eleq2d 2820 |
. . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴)) |
31 | 28, 30 | bitr3d 281 |
. . . . . . 7
⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴)) |
32 | 10 | fveq1d 6894 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)) |
33 | | fmptcof2.6 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) |
34 | | eqidd 2734 |
. . . . . . . 8
⊢ (𝜑 → 𝑤 = 𝑤) |
35 | 32, 33, 34 | breq123d 5163 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)) |
36 | 31, 35 | anbi12d 632 |
. . . . . 6
⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))) |
37 | 4 | nfcri 2891 |
. . . . . . . . . 10
⊢
Ⅎ𝑥 𝑧 ∈ 𝐴 |
38 | | nffvmpt1 6903 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧) |
39 | | fmptcof2.x |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥𝑆 |
40 | 5, 39 | nfmpt 5256 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆) |
41 | | nfcv 2904 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝑤 |
42 | 38, 40, 41 | nfbr 5196 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 |
43 | | nfcsb1v 3919 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇 |
44 | 43 | nfeq2 2921 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇 |
45 | 42, 44 | nfbi 1907 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇) |
46 | 3, 45 | nfim 1900 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
47 | 37, 46 | nfim 1900 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
48 | | eleq1w 2817 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
49 | | fveq2 6892 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)) |
50 | 49 | breq1d 5159 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)) |
51 | | csbeq1a 3908 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇) |
52 | 51 | eqeq2d 2744 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → (𝑤 = 𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
53 | 50, 52 | bibi12d 346 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → ((((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇) ↔ (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
54 | 53 | imbi2d 341 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → ((𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))) |
55 | 48, 54 | imbi12d 345 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))) ↔ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))))) |
56 | | vex 3479 |
. . . . . . . . . . . 12
⊢ 𝑤 ∈ V |
57 | | nfv 1918 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦 𝑅 ∈ 𝐵 |
58 | | fmptcof2.y |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦𝑇 |
59 | 58 | nfeq2 2921 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦 𝑤 = 𝑇 |
60 | 57, 59 | nfan 1903 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇) |
61 | | simpl 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅) |
62 | 61 | eleq1d 2819 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑅 ∈ 𝐵)) |
63 | | simpr 486 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤) |
64 | | fmptcof2.7 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) |
65 | 64 | adantr 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑆 = 𝑇) |
66 | 63, 65 | eqeq12d 2749 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇)) |
67 | 62, 66 | anbi12d 632 |
. . . . . . . . . . . . 13
⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇))) |
68 | | df-mpt 5233 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)} |
69 | 60, 67, 68 | brabgaf 31837 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇))) |
70 | 7, 56, 69 | sylancl 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇))) |
71 | | simpr 486 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
72 | 4 | fvmpt2f 7000 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅) |
73 | 71, 7, 72 | syl2anc 585 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅) |
74 | 73 | breq1d 5159 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)) |
75 | 7 | biantrurd 534 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = 𝑇 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇))) |
76 | 70, 74, 75 | 3bitr4d 311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)) |
77 | 76 | expcom 415 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))) |
78 | 47, 55, 77 | chvarfv 2234 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
79 | 78 | impcom 409 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
80 | 79 | pm5.32da 580 |
. . . . . 6
⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
81 | 36, 80 | bitrd 279 |
. . . . 5
⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
82 | 26, 81 | bitrid 283 |
. . . 4
⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
83 | 21, 82 | bitrd 279 |
. . 3
⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
84 | | vex 3479 |
. . . 4
⊢ 𝑧 ∈ V |
85 | 84, 56 | opelco 5872 |
. . 3
⊢
(⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) |
86 | | df-mpt 5233 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} |
87 | 86 | eleq2i 2826 |
. . . 4
⊢
(⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}) |
88 | 43 | nfeq2 2921 |
. . . . . 6
⊢
Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇 |
89 | 37, 88 | nfan 1903 |
. . . . 5
⊢
Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) |
90 | | nfv 1918 |
. . . . 5
⊢
Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇) |
91 | 51 | eqeq2d 2744 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)) |
92 | 48, 91 | anbi12d 632 |
. . . . 5
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))) |
93 | | eqeq1 2737 |
. . . . . 6
⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
94 | 93 | anbi2d 630 |
. . . . 5
⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))) |
95 | 89, 90, 84, 56, 92, 94 | opelopabf 5546 |
. . . 4
⊢
(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
96 | 87, 95 | bitri 275 |
. . 3
⊢
(⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) |
97 | 83, 85, 96 | 3bitr4g 314 |
. 2
⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇))) |
98 | 1, 2, 97 | eqrelrdv 5793 |
1
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |