Proof of Theorem fmptunsnop
Step | Hyp | Ref
| Expression |
1 | | mptun 6714 |
. 2
⊢ (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)))) |
2 | | fmptunsnop.2 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ 𝐴) |
3 | | difsnid 4814 |
. . . 4
⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴) |
4 | 2, 3 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴) |
5 | 4 | mpteq1d 5242 |
. 2
⊢ (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)))) |
6 | | eldifsni 4794 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ≠ 𝑋) |
7 | 6 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥 ≠ 𝑋) |
8 | 7 | neneqd 2942 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋) |
9 | 8 | iffalsed 4541 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
10 | 9 | mpteq2dva 5247 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥))) |
11 | | fmptunsnop.1 |
. . . . . 6
⊢ (𝜑 → 𝐹 Fn 𝐴) |
12 | | dffn3 6748 |
. . . . . 6
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) |
13 | 11, 12 | sylib 218 |
. . . . 5
⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹) |
14 | | difssd 4146 |
. . . . 5
⊢ (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴) |
15 | 13, 14 | feqresmpt 6977 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥))) |
16 | 10, 15 | eqtr4d 2777 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋}))) |
17 | | iftrue 4536 |
. . . . . 6
⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌) |
18 | 17 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌) |
19 | | fmptunsnop.3 |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
20 | 18, 2, 19 | fmptsnd 7188 |
. . . 4
⊢ (𝜑 → {〈𝑋, 𝑌〉} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)))) |
21 | 20 | eqcomd 2740 |
. . 3
⊢ (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = {〈𝑋, 𝑌〉}) |
22 | 16, 21 | uneq12d 4178 |
. 2
⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) |
23 | 1, 5, 22 | 3eqtr3a 2798 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) |