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Mirrors > Home > ILE Home > Th. List > shftfib | GIF version |
Description: Value of a fiber of the relation 𝐹. (Contributed by Mario Carneiro, 4-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftfib | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shftfval.1 | . . . . . . 7 ⊢ 𝐹 ∈ V | |
2 | 1 | shftfval 10434 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) |
3 | 2 | breqd 3886 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐵(𝐹 shift 𝐴)𝑧 ↔ 𝐵{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑧)) |
4 | vex 2644 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | eleq1 2162 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ)) | |
6 | oveq1 5713 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 − 𝐴) = (𝐵 − 𝐴)) | |
7 | 6 | breq1d 3885 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝑥 − 𝐴)𝐹𝑦 ↔ (𝐵 − 𝐴)𝐹𝑦)) |
8 | 5, 7 | anbi12d 460 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑦))) |
9 | breq2 3879 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐵 − 𝐴)𝐹𝑦 ↔ (𝐵 − 𝐴)𝐹𝑧)) | |
10 | 9 | anbi2d 455 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑦) ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) |
11 | eqid 2100 | . . . . . . 7 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} | |
12 | 8, 10, 11 | brabg 4129 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝑧 ∈ V) → (𝐵{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑧 ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) |
13 | 4, 12 | mpan2 419 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑧 ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) |
14 | 3, 13 | sylan9bb 453 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵(𝐹 shift 𝐴)𝑧 ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) |
15 | ibar 297 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((𝐵 − 𝐴)𝐹𝑧 ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) | |
16 | 15 | adantl 273 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐵 − 𝐴)𝐹𝑧 ↔ (𝐵 ∈ ℂ ∧ (𝐵 − 𝐴)𝐹𝑧))) |
17 | 14, 16 | bitr4d 190 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵(𝐹 shift 𝐴)𝑧 ↔ (𝐵 − 𝐴)𝐹𝑧)) |
18 | 17 | abbidv 2217 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → {𝑧 ∣ 𝐵(𝐹 shift 𝐴)𝑧} = {𝑧 ∣ (𝐵 − 𝐴)𝐹𝑧}) |
19 | imasng 4840 | . . 3 ⊢ (𝐵 ∈ ℂ → ((𝐹 shift 𝐴) “ {𝐵}) = {𝑧 ∣ 𝐵(𝐹 shift 𝐴)𝑧}) | |
20 | 19 | adantl 273 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = {𝑧 ∣ 𝐵(𝐹 shift 𝐴)𝑧}) |
21 | simpr 109 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
22 | simpl 108 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
23 | 21, 22 | subcld 7944 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 − 𝐴) ∈ ℂ) |
24 | imasng 4840 | . . 3 ⊢ ((𝐵 − 𝐴) ∈ ℂ → (𝐹 “ {(𝐵 − 𝐴)}) = {𝑧 ∣ (𝐵 − 𝐴)𝐹𝑧}) | |
25 | 23, 24 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 “ {(𝐵 − 𝐴)}) = {𝑧 ∣ (𝐵 − 𝐴)𝐹𝑧}) |
26 | 18, 20, 25 | 3eqtr4d 2142 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ wcel 1448 {cab 2086 Vcvv 2641 {csn 3474 class class class wbr 3875 {copab 3928 “ cima 4480 (class class class)co 5706 ℂcc 7498 − cmin 7804 shift cshi 10427 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-resscn 7587 ax-1cn 7588 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-sub 7806 df-shft 10428 |
This theorem is referenced by: shftval 10438 |
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