Proof of Theorem imasleval
Step | Hyp | Ref
| Expression |
1 | | fveq2 6437 |
. . . . . . 7
⊢ (𝑐 = 𝑋 → (𝐹‘𝑐) = (𝐹‘𝑋)) |
2 | 1 | breq1d 4885 |
. . . . . 6
⊢ (𝑐 = 𝑋 → ((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑑))) |
3 | | breq1 4878 |
. . . . . 6
⊢ (𝑐 = 𝑋 → (𝑐𝑁𝑑 ↔ 𝑋𝑁𝑑)) |
4 | 2, 3 | bibi12d 337 |
. . . . 5
⊢ (𝑐 = 𝑋 → (((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ 𝑐𝑁𝑑) ↔ ((𝐹‘𝑋) ≤ (𝐹‘𝑑) ↔ 𝑋𝑁𝑑))) |
5 | 4 | imbi2d 332 |
. . . 4
⊢ (𝑐 = 𝑋 → ((𝜑 → ((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ 𝑐𝑁𝑑)) ↔ (𝜑 → ((𝐹‘𝑋) ≤ (𝐹‘𝑑) ↔ 𝑋𝑁𝑑)))) |
6 | | fveq2 6437 |
. . . . . . 7
⊢ (𝑑 = 𝑌 → (𝐹‘𝑑) = (𝐹‘𝑌)) |
7 | 6 | breq2d 4887 |
. . . . . 6
⊢ (𝑑 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑑) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
8 | | breq2 4879 |
. . . . . 6
⊢ (𝑑 = 𝑌 → (𝑋𝑁𝑑 ↔ 𝑋𝑁𝑌)) |
9 | 7, 8 | bibi12d 337 |
. . . . 5
⊢ (𝑑 = 𝑌 → (((𝐹‘𝑋) ≤ (𝐹‘𝑑) ↔ 𝑋𝑁𝑑) ↔ ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌))) |
10 | 9 | imbi2d 332 |
. . . 4
⊢ (𝑑 = 𝑌 → ((𝜑 → ((𝐹‘𝑋) ≤ (𝐹‘𝑑) ↔ 𝑋𝑁𝑑)) ↔ (𝜑 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌)))) |
11 | | imasless.f |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
12 | | fofn 6359 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) |
13 | 11, 12 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn 𝑉) |
14 | 13 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝐹 Fn 𝑉) |
15 | | fndm 6227 |
. . . . . . . . . 10
⊢ (𝐹 Fn 𝑉 → dom 𝐹 = 𝑉) |
16 | 14, 15 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → dom 𝐹 = 𝑉) |
17 | 16 | rexeqdv 3357 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (∃𝑎 ∈ dom 𝐹(𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ∃𝑎 ∈ 𝑉 (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
18 | | fnbrfvb 6486 |
. . . . . . . . . . . 12
⊢ ((𝐹 Fn 𝑉 ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑎) = (𝐹‘𝑐) ↔ 𝑎𝐹(𝐹‘𝑐))) |
19 | 14, 18 | sylan 575 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑎) = (𝐹‘𝑐) ↔ 𝑎𝐹(𝐹‘𝑐))) |
20 | 19 | anbi1d 623 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
21 | | ancom 454 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎𝑁𝑏 ∧ 𝑏𝐹(𝐹‘𝑑)) ↔ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏)) |
22 | | vex 3417 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑏 ∈ V |
23 | | fvex 6450 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹‘𝑑) ∈ V |
24 | 22, 23 | breldm 5565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏𝐹(𝐹‘𝑑) → 𝑏 ∈ dom 𝐹) |
25 | 24 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) → 𝑏 ∈ dom 𝐹) |
26 | 25 | pm4.71ri 556 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ (𝑏 ∈ dom 𝐹 ∧ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
27 | 21, 26 | bitri 267 |
. . . . . . . . . . . . . 14
⊢ ((𝑎𝑁𝑏 ∧ 𝑏𝐹(𝐹‘𝑑)) ↔ (𝑏 ∈ dom 𝐹 ∧ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
28 | 27 | exbii 1947 |
. . . . . . . . . . . . 13
⊢
(∃𝑏(𝑎𝑁𝑏 ∧ 𝑏𝐹(𝐹‘𝑑)) ↔ ∃𝑏(𝑏 ∈ dom 𝐹 ∧ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
29 | | vex 3417 |
. . . . . . . . . . . . . 14
⊢ 𝑎 ∈ V |
30 | 29, 23 | brco 5529 |
. . . . . . . . . . . . 13
⊢ (𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑) ↔ ∃𝑏(𝑎𝑁𝑏 ∧ 𝑏𝐹(𝐹‘𝑑))) |
31 | | df-rex 3123 |
. . . . . . . . . . . . 13
⊢
(∃𝑏 ∈ dom
𝐹(𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ∃𝑏(𝑏 ∈ dom 𝐹 ∧ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
32 | 28, 30, 31 | 3bitr4i 295 |
. . . . . . . . . . . 12
⊢ (𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑) ↔ ∃𝑏 ∈ dom 𝐹(𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏)) |
33 | 14 | ad2antrr 717 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → 𝐹 Fn 𝑉) |
34 | | fnbrfvb 6486 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹 Fn 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ 𝑏𝐹(𝐹‘𝑑))) |
35 | 33, 34 | sylan 575 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ 𝑏𝐹(𝐹‘𝑑))) |
36 | 35 | anbi1d 623 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) ∧ 𝑏 ∈ 𝑉) → (((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
37 | | imasleval.e |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑))) |
38 | 37 | 3expa 1151 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑))) |
39 | 38 | an32s 642 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑))) |
40 | 39 | anassrs 461 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑))) |
41 | 40 | impl 449 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑)) |
42 | 41 | pm5.32da 574 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
43 | 42 | an32s 642 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) ∧ 𝑏 ∈ 𝑉) → (((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
44 | 36, 43 | bitr3d 273 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) ∧ 𝑏 ∈ 𝑉) → ((𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
45 | 44 | rexbidva 3259 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (∃𝑏 ∈ 𝑉 (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ∃𝑏 ∈ 𝑉 ((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
46 | | r19.41v 3299 |
. . . . . . . . . . . . . 14
⊢
(∃𝑏 ∈
𝑉 ((𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑) ↔ (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑)) |
47 | 45, 46 | syl6bb 279 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (∃𝑏 ∈ 𝑉 (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
48 | 16 | rexeqdv 3357 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (∃𝑏 ∈ dom 𝐹(𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ∃𝑏 ∈ 𝑉 (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
49 | 48 | ad2antrr 717 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (∃𝑏 ∈ dom 𝐹(𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ ∃𝑏 ∈ 𝑉 (𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏))) |
50 | | simprr 789 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑑 ∈ 𝑉) |
51 | | eqid 2825 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹‘𝑑) = (𝐹‘𝑑) |
52 | | fveqeq2 6446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑑 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑑) = (𝐹‘𝑑))) |
53 | 52 | rspcev 3526 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑑 ∈ 𝑉 ∧ (𝐹‘𝑑) = (𝐹‘𝑑)) → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑)) |
54 | 50, 51, 53 | sylancl 580 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑)) |
55 | 54 | biantrurd 528 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐𝑁𝑑 ↔ (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
56 | 55 | ad2antrr 717 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (𝑐𝑁𝑑 ↔ (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = (𝐹‘𝑑) ∧ 𝑐𝑁𝑑))) |
57 | 47, 49, 56 | 3bitr4d 303 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (∃𝑏 ∈ dom 𝐹(𝑏𝐹(𝐹‘𝑑) ∧ 𝑎𝑁𝑏) ↔ 𝑐𝑁𝑑)) |
58 | 32, 57 | syl5bb 275 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ (𝐹‘𝑎) = (𝐹‘𝑐)) → (𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑) ↔ 𝑐𝑁𝑑)) |
59 | 58 | pm5.32da 574 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
60 | 20, 59 | bitr3d 273 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ 𝑎 ∈ 𝑉) → ((𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
61 | 60 | rexbidva 3259 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (∃𝑎 ∈ 𝑉 (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ∃𝑎 ∈ 𝑉 ((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
62 | 17, 61 | bitrd 271 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (∃𝑎 ∈ dom 𝐹(𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ∃𝑎 ∈ 𝑉 ((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
63 | | fvex 6450 |
. . . . . . . . . . . 12
⊢ (𝐹‘𝑐) ∈ V |
64 | 63, 29 | brcnv 5541 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑐)◡𝐹𝑎 ↔ 𝑎𝐹(𝐹‘𝑐)) |
65 | 64 | anbi1i 617 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑐)◡𝐹𝑎 ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑))) |
66 | 29, 63 | breldm 5565 |
. . . . . . . . . . . 12
⊢ (𝑎𝐹(𝐹‘𝑐) → 𝑎 ∈ dom 𝐹) |
67 | 66 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) → 𝑎 ∈ dom 𝐹) |
68 | 67 | pm4.71ri 556 |
. . . . . . . . . 10
⊢ ((𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ (𝑎 ∈ dom 𝐹 ∧ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
69 | 65, 68 | bitri 267 |
. . . . . . . . 9
⊢ (((𝐹‘𝑐)◡𝐹𝑎 ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ (𝑎 ∈ dom 𝐹 ∧ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
70 | 69 | exbii 1947 |
. . . . . . . 8
⊢
(∃𝑎((𝐹‘𝑐)◡𝐹𝑎 ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ∃𝑎(𝑎 ∈ dom 𝐹 ∧ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
71 | 63, 23 | brco 5529 |
. . . . . . . 8
⊢ ((𝐹‘𝑐)((𝐹 ∘ 𝑁) ∘ ◡𝐹)(𝐹‘𝑑) ↔ ∃𝑎((𝐹‘𝑐)◡𝐹𝑎 ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑))) |
72 | | df-rex 3123 |
. . . . . . . 8
⊢
(∃𝑎 ∈ dom
𝐹(𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ ∃𝑎(𝑎 ∈ dom 𝐹 ∧ (𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)))) |
73 | 70, 71, 72 | 3bitr4ri 296 |
. . . . . . 7
⊢
(∃𝑎 ∈ dom
𝐹(𝑎𝐹(𝐹‘𝑐) ∧ 𝑎(𝐹 ∘ 𝑁)(𝐹‘𝑑)) ↔ (𝐹‘𝑐)((𝐹 ∘ 𝑁) ∘ ◡𝐹)(𝐹‘𝑑)) |
74 | | r19.41v 3299 |
. . . . . . 7
⊢
(∃𝑎 ∈
𝑉 ((𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑) ↔ (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑)) |
75 | 62, 73, 74 | 3bitr3g 305 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝐹‘𝑐)((𝐹 ∘ 𝑁) ∘ ◡𝐹)(𝐹‘𝑑) ↔ (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
76 | | imasless.u |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
77 | | imasless.v |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
78 | | imasless.r |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ 𝑍) |
79 | | imasleval.n |
. . . . . . . . 9
⊢ 𝑁 = (le‘𝑅) |
80 | | imasless.l |
. . . . . . . . 9
⊢ ≤ =
(le‘𝑈) |
81 | 76, 77, 11, 78, 79, 80 | imasle 16543 |
. . . . . . . 8
⊢ (𝜑 → ≤ = ((𝐹 ∘ 𝑁) ∘ ◡𝐹)) |
82 | 81 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ≤ = ((𝐹 ∘ 𝑁) ∘ ◡𝐹)) |
83 | 82 | breqd 4886 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ (𝐹‘𝑐)((𝐹 ∘ 𝑁) ∘ ◡𝐹)(𝐹‘𝑑))) |
84 | | simprl 787 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑐 ∈ 𝑉) |
85 | | eqid 2825 |
. . . . . . . 8
⊢ (𝐹‘𝑐) = (𝐹‘𝑐) |
86 | | fveqeq2 6446 |
. . . . . . . . 9
⊢ (𝑎 = 𝑐 → ((𝐹‘𝑎) = (𝐹‘𝑐) ↔ (𝐹‘𝑐) = (𝐹‘𝑐))) |
87 | 86 | rspcev 3526 |
. . . . . . . 8
⊢ ((𝑐 ∈ 𝑉 ∧ (𝐹‘𝑐) = (𝐹‘𝑐)) → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = (𝐹‘𝑐)) |
88 | 84, 85, 87 | sylancl 580 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = (𝐹‘𝑐)) |
89 | 88 | biantrurd 528 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐𝑁𝑑 ↔ (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = (𝐹‘𝑐) ∧ 𝑐𝑁𝑑))) |
90 | 75, 83, 89 | 3bitr4d 303 |
. . . . 5
⊢ ((𝜑 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ 𝑐𝑁𝑑)) |
91 | 90 | expcom 404 |
. . . 4
⊢ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (𝜑 → ((𝐹‘𝑐) ≤ (𝐹‘𝑑) ↔ 𝑐𝑁𝑑))) |
92 | 5, 10, 91 | vtocl2ga 3491 |
. . 3
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝜑 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌))) |
93 | 92 | com12 32 |
. 2
⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌))) |
94 | 93 | 3impib 1148 |
1
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌)) |