| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | relopabv 5830 | . . . . 5
⊢ Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} | 
| 2 | 1 | a1i 11 | . . . 4
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) | 
| 3 |  | fnfun 6667 | . . . . . 6
⊢ (𝐹 Fn 𝐵 → Fun 𝐹) | 
| 4 | 3 | adantr 480 | . . . . 5
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun 𝐹) | 
| 5 |  | funmo 6580 | . . . . . . 7
⊢ (Fun
𝐹 → ∃*𝑤(𝑧 − 𝐴)𝐹𝑤) | 
| 6 |  | vex 3483 | . . . . . . . . . 10
⊢ 𝑧 ∈ V | 
| 7 |  | vex 3483 | . . . . . . . . . 10
⊢ 𝑤 ∈ V | 
| 8 |  | eleq1w 2823 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ)) | 
| 9 |  | oveq1 7439 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → (𝑥 − 𝐴) = (𝑧 − 𝐴)) | 
| 10 | 9 | breq1d 5152 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → ((𝑥 − 𝐴)𝐹𝑦 ↔ (𝑧 − 𝐴)𝐹𝑦)) | 
| 11 | 8, 10 | anbi12d 632 | . . . . . . . . . 10
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑦))) | 
| 12 |  | breq2 5146 | . . . . . . . . . . 11
⊢ (𝑦 = 𝑤 → ((𝑧 − 𝐴)𝐹𝑦 ↔ (𝑧 − 𝐴)𝐹𝑤)) | 
| 13 | 12 | anbi2d 630 | . . . . . . . . . 10
⊢ (𝑦 = 𝑤 → ((𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤))) | 
| 14 |  | eqid 2736 | . . . . . . . . . 10
⊢
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} | 
| 15 | 6, 7, 11, 13, 14 | brab 5547 | . . . . . . . . 9
⊢ (𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤 ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)) | 
| 16 | 15 | simprbi 496 | . . . . . . . 8
⊢ (𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤 → (𝑧 − 𝐴)𝐹𝑤) | 
| 17 | 16 | moimi 2544 | . . . . . . 7
⊢
(∃*𝑤(𝑧 − 𝐴)𝐹𝑤 → ∃*𝑤 𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤) | 
| 18 | 5, 17 | syl 17 | . . . . . 6
⊢ (Fun
𝐹 → ∃*𝑤 𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤) | 
| 19 | 18 | alrimiv 1926 | . . . . 5
⊢ (Fun
𝐹 → ∀𝑧∃*𝑤 𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤) | 
| 20 | 4, 19 | syl 17 | . . . 4
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → ∀𝑧∃*𝑤 𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤) | 
| 21 |  | dffun6 6573 | . . . 4
⊢ (Fun
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ↔ (Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∧ ∀𝑧∃*𝑤 𝑧{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤)) | 
| 22 | 2, 20, 21 | sylanbrc 583 | . . 3
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) | 
| 23 |  | shftfval.1 | . . . . . 6
⊢ 𝐹 ∈ V | 
| 24 | 23 | shftfval 15110 | . . . . 5
⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) | 
| 25 | 24 | adantl 481 | . . . 4
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) | 
| 26 | 25 | funeqd 6587 | . . 3
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (Fun (𝐹 shift 𝐴) ↔ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})) | 
| 27 | 22, 26 | mpbird 257 | . 2
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun (𝐹 shift 𝐴)) | 
| 28 | 23 | shftdm 15111 | . . 3
⊢ (𝐴 ∈ ℂ → dom
(𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹}) | 
| 29 |  | fndm 6670 | . . . . 5
⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵) | 
| 30 | 29 | eleq2d 2826 | . . . 4
⊢ (𝐹 Fn 𝐵 → ((𝑥 − 𝐴) ∈ dom 𝐹 ↔ (𝑥 − 𝐴) ∈ 𝐵)) | 
| 31 | 30 | rabbidv 3443 | . . 3
⊢ (𝐹 Fn 𝐵 → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹} = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) | 
| 32 | 28, 31 | sylan9eqr 2798 | . 2
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) | 
| 33 |  | df-fn 6563 | . 2
⊢ ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ↔ (Fun (𝐹 shift 𝐴) ∧ dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵})) | 
| 34 | 27, 32, 33 | sylanbrc 583 | 1
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |