Proof of Theorem fvmptt
Step | Hyp | Ref
| Expression |
1 | | simp2 1136 |
. . 3
⊢
((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
2 | 1 | fveq1d 6773 |
. 2
⊢
((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
3 | | risset 3196 |
. . . . 5
⊢ (𝐴 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐷 𝑥 = 𝐴) |
4 | | elex 3449 |
. . . . . 6
⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) |
5 | | nfa1 2152 |
. . . . . . 7
⊢
Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) |
6 | | nfv 1921 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝐶 ∈ V |
7 | | nffvmpt1 6782 |
. . . . . . . . 9
⊢
Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) |
8 | 7 | nfeq1 2924 |
. . . . . . . 8
⊢
Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶 |
9 | 6, 8 | nfim 1903 |
. . . . . . 7
⊢
Ⅎ𝑥(𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
10 | | simprl 768 |
. . . . . . . . . . . . 13
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝑥 ∈ 𝐷) |
11 | | simplr 766 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶) |
12 | | simprr 770 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐶 ∈ V) |
13 | 11, 12 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 ∈ V) |
14 | | eqid 2740 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) |
15 | 14 | fvmpt2 6883 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
16 | 10, 13, 15 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
17 | | simpll 764 |
. . . . . . . . . . . . 13
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝑥 = 𝐴) |
18 | 17 | fveq2d 6775 |
. . . . . . . . . . . 12
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
19 | 16, 18, 11 | 3eqtr3d 2788 |
. . . . . . . . . . 11
⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
20 | 19 | exp43 437 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 → (𝑥 ∈ 𝐷 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))) |
21 | 20 | a2i 14 |
. . . . . . . . 9
⊢ ((𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐷 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))) |
22 | 21 | com23 86 |
. . . . . . . 8
⊢ ((𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))) |
23 | 22 | sps 2182 |
. . . . . . 7
⊢
(∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))) |
24 | 5, 9, 23 | rexlimd 3248 |
. . . . . 6
⊢
(∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (∃𝑥 ∈ 𝐷 𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))) |
25 | 4, 24 | syl7 74 |
. . . . 5
⊢
(∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (∃𝑥 ∈ 𝐷 𝑥 = 𝐴 → (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))) |
26 | 3, 25 | syl5bi 241 |
. . . 4
⊢
(∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))) |
27 | 26 | imp32 419 |
. . 3
⊢
((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
28 | 27 | 3adant2 1130 |
. 2
⊢
((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
29 | 2, 28 | eqtrd 2780 |
1
⊢
((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶) |