Step | Hyp | Ref
| Expression |
1 | | wemaplem2.px1 |
. . . 4
⊢ (𝜑 → 𝑎 ∈ 𝐴) |
2 | | wemaplem2.xq1 |
. . . 4
⊢ (𝜑 → 𝑏 ∈ 𝐴) |
3 | 1, 2 | ifcld 4081 |
. . 3
⊢ (𝜑 → if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴) |
4 | | wemaplem2.px2 |
. . . . . . 7
⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) |
5 | 4 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) |
6 | | wemaplem2.xq3 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) |
7 | | breq1 4586 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑎 → (𝑐𝑅𝑏 ↔ 𝑎𝑅𝑏)) |
8 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑎 → (𝑋‘𝑐) = (𝑋‘𝑎)) |
9 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑎 → (𝑄‘𝑐) = (𝑄‘𝑎)) |
10 | 8, 9 | eqeq12d 2625 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑎 → ((𝑋‘𝑐) = (𝑄‘𝑐) ↔ (𝑋‘𝑎) = (𝑄‘𝑎))) |
11 | 7, 10 | imbi12d 333 |
. . . . . . . . 9
⊢ (𝑐 = 𝑎 → ((𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)) ↔ (𝑎𝑅𝑏 → (𝑋‘𝑎) = (𝑄‘𝑎)))) |
12 | 11 | rspcva 3280 |
. . . . . . . 8
⊢ ((𝑎 ∈ 𝐴 ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → (𝑎𝑅𝑏 → (𝑋‘𝑎) = (𝑄‘𝑎))) |
13 | 1, 6, 12 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝑎𝑅𝑏 → (𝑋‘𝑎) = (𝑄‘𝑎))) |
14 | 13 | imp 444 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑋‘𝑎) = (𝑄‘𝑎)) |
15 | 5, 14 | breqtrd 4609 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘𝑎)𝑆(𝑄‘𝑎)) |
16 | | iftrue 4042 |
. . . . . . . 8
⊢ (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎) |
17 | 16 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑎)) |
18 | 16 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑎)) |
19 | 17, 18 | breq12d 4596 |
. . . . . 6
⊢ (𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎))) |
20 | 19 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎))) |
21 | 15, 20 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
22 | | wemaplem2.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 Po 𝐵) |
23 | 22 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → 𝑆 Po 𝐵) |
24 | | wemaplem2.p |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑𝑚 𝐴)) |
25 | | elmapi 7765 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (𝐵 ↑𝑚 𝐴) → 𝑃:𝐴⟶𝐵) |
26 | 24, 25 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃:𝐴⟶𝐵) |
27 | 26, 2 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (𝜑 → (𝑃‘𝑏) ∈ 𝐵) |
28 | | wemaplem2.x |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 𝐴)) |
29 | | elmapi 7765 |
. . . . . . . . . 10
⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐴) → 𝑋:𝐴⟶𝐵) |
30 | 28, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋:𝐴⟶𝐵) |
31 | 30, 2 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (𝜑 → (𝑋‘𝑏) ∈ 𝐵) |
32 | | wemaplem2.q |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑𝑚 𝐴)) |
33 | | elmapi 7765 |
. . . . . . . . . 10
⊢ (𝑄 ∈ (𝐵 ↑𝑚 𝐴) → 𝑄:𝐴⟶𝐵) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄:𝐴⟶𝐵) |
35 | 34, 2 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘𝑏) ∈ 𝐵) |
36 | 27, 31, 35 | 3jca 1235 |
. . . . . . 7
⊢ (𝜑 → ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) |
37 | 36 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) |
38 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → (𝑃‘𝑎) = (𝑃‘𝑏)) |
39 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏)) |
40 | 38, 39 | breq12d 4596 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ↔ (𝑃‘𝑏)𝑆(𝑋‘𝑏))) |
41 | 4, 40 | syl5ibcom 234 |
. . . . . . 7
⊢ (𝜑 → (𝑎 = 𝑏 → (𝑃‘𝑏)𝑆(𝑋‘𝑏))) |
42 | 41 | imp 444 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘𝑏)𝑆(𝑋‘𝑏)) |
43 | | wemaplem2.xq2 |
. . . . . . 7
⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) |
44 | 43 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) |
45 | | potr 4971 |
. . . . . . 7
⊢ ((𝑆 Po 𝐵 ∧ ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) → (((𝑃‘𝑏)𝑆(𝑋‘𝑏) ∧ (𝑋‘𝑏)𝑆(𝑄‘𝑏)) → (𝑃‘𝑏)𝑆(𝑄‘𝑏))) |
46 | 45 | imp 444 |
. . . . . 6
⊢ (((𝑆 Po 𝐵 ∧ ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) ∧ ((𝑃‘𝑏)𝑆(𝑋‘𝑏) ∧ (𝑋‘𝑏)𝑆(𝑄‘𝑏))) → (𝑃‘𝑏)𝑆(𝑄‘𝑏)) |
47 | 23, 37, 42, 44, 46 | syl22anc 1319 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘𝑏)𝑆(𝑄‘𝑏)) |
48 | | ifeq1 4040 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = if(𝑎𝑅𝑏, 𝑏, 𝑏)) |
49 | | ifid 4075 |
. . . . . . . . 9
⊢ if(𝑎𝑅𝑏, 𝑏, 𝑏) = 𝑏 |
50 | 48, 49 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏) |
51 | 50 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑏)) |
52 | 50 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑏)) |
53 | 51, 52 | breq12d 4596 |
. . . . . 6
⊢ (𝑎 = 𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏))) |
54 | 53 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏))) |
55 | 47, 54 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
56 | | wemaplem2.px3 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) |
57 | | breq1 4586 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑏 → (𝑐𝑅𝑎 ↔ 𝑏𝑅𝑎)) |
58 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑏 → (𝑃‘𝑐) = (𝑃‘𝑏)) |
59 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑏 → (𝑋‘𝑐) = (𝑋‘𝑏)) |
60 | 58, 59 | eqeq12d 2625 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑏 → ((𝑃‘𝑐) = (𝑋‘𝑐) ↔ (𝑃‘𝑏) = (𝑋‘𝑏))) |
61 | 57, 60 | imbi12d 333 |
. . . . . . . . 9
⊢ (𝑐 = 𝑏 → ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ↔ (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑋‘𝑏)))) |
62 | 61 | rspcva 3280 |
. . . . . . . 8
⊢ ((𝑏 ∈ 𝐴 ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) → (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑋‘𝑏))) |
63 | 2, 56, 62 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑋‘𝑏))) |
64 | 63 | imp 444 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘𝑏) = (𝑋‘𝑏)) |
65 | 43 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) |
66 | 64, 65 | eqbrtrd 4605 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘𝑏)𝑆(𝑄‘𝑏)) |
67 | | wemaplem2.r |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 Or 𝐴) |
68 | | sopo 4976 |
. . . . . . . . 9
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) |
69 | 67, 68 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 Po 𝐴) |
70 | | po2nr 4972 |
. . . . . . . 8
⊢ ((𝑅 Po 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴)) → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏)) |
71 | 69, 2, 1, 70 | syl12anc 1316 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏)) |
72 | | nan 602 |
. . . . . . 7
⊢ ((𝜑 → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏)) ↔ ((𝜑 ∧ 𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏)) |
73 | 71, 72 | mpbi 219 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏) |
74 | | iffalse 4045 |
. . . . . . . 8
⊢ (¬
𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏) |
75 | 74 | fveq2d 6107 |
. . . . . . 7
⊢ (¬
𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑏)) |
76 | 74 | fveq2d 6107 |
. . . . . . 7
⊢ (¬
𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑏)) |
77 | 75, 76 | breq12d 4596 |
. . . . . 6
⊢ (¬
𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏))) |
78 | 73, 77 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏))) |
79 | 66, 78 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
80 | | solin 4982 |
. . . . 5
⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) |
81 | 67, 1, 2, 80 | syl12anc 1316 |
. . . 4
⊢ (𝜑 → (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) |
82 | 21, 55, 79, 81 | mpjao3dan 1387 |
. . 3
⊢ (𝜑 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
83 | | r19.26 3046 |
. . . . 5
⊢
(∀𝑐 ∈
𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) ↔ (∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))) |
84 | 56, 6, 83 | sylanbrc 695 |
. . . 4
⊢ (𝜑 → ∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))) |
85 | 67, 1, 2 | 3jca 1235 |
. . . . 5
⊢ (𝜑 → (𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) |
86 | | prth 593 |
. . . . . . 7
⊢ (((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → ((𝑃‘𝑐) = (𝑋‘𝑐) ∧ (𝑋‘𝑐) = (𝑄‘𝑐)))) |
87 | | eqtr 2629 |
. . . . . . 7
⊢ (((𝑃‘𝑐) = (𝑋‘𝑐) ∧ (𝑋‘𝑐) = (𝑄‘𝑐)) → (𝑃‘𝑐) = (𝑄‘𝑐)) |
88 | 86, 87 | syl6 34 |
. . . . . 6
⊢ (((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))) |
89 | 88 | ralimi 2936 |
. . . . 5
⊢
(∀𝑐 ∈
𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))) |
90 | | simpl1 1057 |
. . . . . . . . 9
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑅 Or 𝐴) |
91 | | simpr 476 |
. . . . . . . . 9
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ 𝐴) |
92 | | simpl2 1058 |
. . . . . . . . 9
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑎 ∈ 𝐴) |
93 | | simpl3 1059 |
. . . . . . . . 9
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ 𝐴) |
94 | | soltmin 5451 |
. . . . . . . . 9
⊢ ((𝑅 Or 𝐴 ∧ (𝑐 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏))) |
95 | 90, 91, 92, 93, 94 | syl13anc 1320 |
. . . . . . . 8
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏))) |
96 | 95 | biimpd 218 |
. . . . . . 7
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏))) |
97 | 96 | imim1d 80 |
. . . . . 6
⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
98 | 97 | ralimdva 2945 |
. . . . 5
⊢ ((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)) → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
99 | 85, 89, 98 | syl2im 39 |
. . . 4
⊢ (𝜑 → (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
100 | 84, 99 | mpd 15 |
. . 3
⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))) |
101 | | fveq2 6103 |
. . . . . 6
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑑) = (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
102 | | fveq2 6103 |
. . . . . 6
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑄‘𝑑) = (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
103 | 101, 102 | breq12d 4596 |
. . . . 5
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ↔ (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))) |
104 | | breq2 4587 |
. . . . . . 7
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑑 ↔ 𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏))) |
105 | 104 | imbi1d 330 |
. . . . . 6
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)) ↔ (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
106 | 105 | ralbidv 2969 |
. . . . 5
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)) ↔ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
107 | 103, 106 | anbi12d 743 |
. . . 4
⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))) ↔ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))) |
108 | 107 | rspcev 3282 |
. . 3
⊢
((if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴 ∧ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) → ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
109 | 3, 82, 100, 108 | syl12anc 1316 |
. 2
⊢ (𝜑 → ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)))) |
110 | | wemapso.t |
. . . 4
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
111 | 110 | wemaplem1 8334 |
. . 3
⊢ ((𝑃 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑄 ∈ (𝐵 ↑𝑚 𝐴)) → (𝑃𝑇𝑄 ↔ ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))))) |
112 | 24, 32, 111 | syl2anc 691 |
. 2
⊢ (𝜑 → (𝑃𝑇𝑄 ↔ ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))))) |
113 | 109, 112 | mpbird 246 |
1
⊢ (𝜑 → 𝑃𝑇𝑄) |