Proof of Theorem elsetpreimafvbi
Step | Hyp | Ref
| Expression |
1 | | fniniseg 6837 |
. . . . . 6
⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)))) |
2 | | fniniseg 6837 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)))) |
3 | | eqeq2 2750 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹‘𝑌) = (𝐹‘𝑥) ↔ (𝐹‘𝑌) = (𝐹‘𝑋))) |
4 | 3 | anbi2d 632 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) |
5 | 4 | eqcoms 2746 |
. . . . . . . . 9
⊢ ((𝐹‘𝑋) = (𝐹‘𝑥) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) |
6 | 2, 5 | sylan9bb 513 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) |
7 | 6 | ex 416 |
. . . . . . 7
⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑋) = (𝐹‘𝑥) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))) |
8 | 7 | adantld 494 |
. . . . . 6
⊢ (𝐹 Fn 𝐴 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))) |
9 | 1, 8 | sylbid 243 |
. . . . 5
⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))) |
10 | | eleq2 2821 |
. . . . . 6
⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))) |
11 | | eleq2 2821 |
. . . . . . 7
⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ 𝑆 ↔ 𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))) |
12 | 11 | bibi1d 347 |
. . . . . 6
⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ((𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))) ↔ (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))) |
13 | 10, 12 | imbi12d 348 |
. . . . 5
⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ((𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) ↔ (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))) |
14 | 9, 13 | syl5ibr 249 |
. . . 4
⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))) |
15 | 14 | rexlimivw 3192 |
. . 3
⊢
(∃𝑥 ∈
𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))) |
16 | | setpreimafvex.p |
. . . 4
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
17 | 16 | elsetpreimafv 44371 |
. . 3
⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) |
18 | 15, 17 | syl11 33 |
. 2
⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))) |
19 | 18 | 3imp 1112 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) |