Step | Hyp | Ref
| Expression |
1 | | funfvop 6822 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) → 〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹) |
2 | | simplll 773 |
. . . . . . . 8
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Fun 𝐹) |
3 | | funrel 6374 |
. . . . . . . 8
⊢ (Fun
𝐹 → Rel 𝐹) |
4 | 2, 3 | syl 17 |
. . . . . . 7
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Rel 𝐹) |
5 | | simplr 767 |
. . . . . . 7
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 ∈ 𝐹) |
6 | | 1st2nd 7740 |
. . . . . . 7
⊢ ((Rel
𝐹 ∧ 𝑝 ∈ 𝐹) → 𝑝 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
7 | 4, 5, 6 | syl2anc 586 |
. . . . . 6
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
8 | | simpr 487 |
. . . . . . 7
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 = (1st ‘𝑝)) |
9 | | simpllr 774 |
. . . . . . . 8
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 ∈ dom 𝐹) |
10 | 8 | opeq1d 4811 |
. . . . . . . . . 10
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 〈𝑋, (2nd ‘𝑝)〉 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
11 | 7, 10 | eqtr4d 2861 |
. . . . . . . . 9
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = 〈𝑋, (2nd ‘𝑝)〉) |
12 | 11, 5 | eqeltrrd 2916 |
. . . . . . . 8
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 〈𝑋, (2nd ‘𝑝)〉 ∈ 𝐹) |
13 | | funopfvb 6723 |
. . . . . . . . 9
⊢ ((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹‘𝑋) = (2nd ‘𝑝) ↔ 〈𝑋, (2nd ‘𝑝)〉 ∈ 𝐹)) |
14 | 13 | biimpar 480 |
. . . . . . . 8
⊢ (((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 〈𝑋, (2nd ‘𝑝)〉 ∈ 𝐹) → (𝐹‘𝑋) = (2nd ‘𝑝)) |
15 | 2, 9, 12, 14 | syl21anc 835 |
. . . . . . 7
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → (𝐹‘𝑋) = (2nd ‘𝑝)) |
16 | 8, 15 | opeq12d 4813 |
. . . . . 6
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 〈𝑋, (𝐹‘𝑋)〉 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
17 | 7, 16 | eqtr4d 2861 |
. . . . 5
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) |
18 | | simpr 487 |
. . . . . . 7
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) → 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) |
19 | 18 | fveq2d 6676 |
. . . . . 6
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) → (1st ‘𝑝) = (1st
‘〈𝑋, (𝐹‘𝑋)〉)) |
20 | | fvex 6685 |
. . . . . . . 8
⊢ (𝐹‘𝑋) ∈ V |
21 | | op1stg 7703 |
. . . . . . . 8
⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st
‘〈𝑋, (𝐹‘𝑋)〉) = 𝑋) |
22 | 20, 21 | mpan2 689 |
. . . . . . 7
⊢ (𝑋 ∈ dom 𝐹 → (1st ‘〈𝑋, (𝐹‘𝑋)〉) = 𝑋) |
23 | 22 | ad3antlr 729 |
. . . . . 6
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) → (1st
‘〈𝑋, (𝐹‘𝑋)〉) = 𝑋) |
24 | 19, 23 | eqtr2d 2859 |
. . . . 5
⊢ ((((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉) → 𝑋 = (1st ‘𝑝)) |
25 | 17, 24 | impbida 799 |
. . . 4
⊢ (((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) → (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉)) |
26 | 25 | ralrimiva 3184 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉)) |
27 | | eqeq2 2835 |
. . . . . 6
⊢ (𝑞 = 〈𝑋, (𝐹‘𝑋)〉 → (𝑝 = 𝑞 ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉)) |
28 | 27 | bibi2d 345 |
. . . . 5
⊢ (𝑞 = 〈𝑋, (𝐹‘𝑋)〉 → ((𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉))) |
29 | 28 | ralbidv 3199 |
. . . 4
⊢ (𝑞 = 〈𝑋, (𝐹‘𝑋)〉 → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉))) |
30 | 29 | rspcev 3625 |
. . 3
⊢
((〈𝑋, (𝐹‘𝑋)〉 ∈ 𝐹 ∧ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 〈𝑋, (𝐹‘𝑋)〉)) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞)) |
31 | 1, 26, 30 | syl2anc 586 |
. 2
⊢ ((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞)) |
32 | | reu6 3719 |
. 2
⊢
(∃!𝑝 ∈
𝐹 𝑋 = (1st ‘𝑝) ↔ ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞)) |
33 | 31, 32 | sylibr 236 |
1
⊢ ((Fun
𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝)) |