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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fargshiftf | Structured version Visualization version GIF version |
Description: If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
Ref | Expression |
---|---|
fargshift.g | ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) |
Ref | Expression |
---|---|
fargshiftf | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6487 | . . . 4 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁)) | |
2 | fseq1hash 13733 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁) | |
3 | 1, 2 | sylan2 595 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁) |
4 | eleq1 2877 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)) | |
5 | oveq2 7143 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (1...𝑁) = (1...(♯‘𝐹))) | |
6 | 5 | feq2d 6473 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝐹:(1...𝑁)⟶dom 𝐸 ↔ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)) |
7 | 4, 6 | anbi12d 633 | . . . . 5 ⊢ (𝑁 = (♯‘𝐹) → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))) |
8 | 7 | eqcoms 2806 | . . . 4 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))) |
9 | fz0add1fz1 13102 | . . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝑥 + 1) ∈ (1...(♯‘𝐹))) | |
10 | ffvelrn 6826 | . . . . . . . 8 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ (𝑥 + 1) ∈ (1...(♯‘𝐹))) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸) | |
11 | 10 | expcom 417 | . . . . . . 7 ⊢ ((𝑥 + 1) ∈ (1...(♯‘𝐹)) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
12 | 9, 11 | syl 17 | . . . . . 6 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
13 | 12 | impancom 455 | . . . . 5 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
14 | 13 | ralrimiv 3148 | . . . 4 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸) |
15 | 8, 14 | syl6bi 256 | . . 3 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
16 | 3, 15 | mpcom 38 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸) |
17 | fargshift.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) | |
18 | 17 | fmpt 6851 | . 2 ⊢ (∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸 ↔ 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
19 | 16, 18 | sylib 221 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 dom cdm 5519 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 ℕ0cn0 11885 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 |
This theorem is referenced by: fargshiftf1 43958 fargshiftfo 43959 |
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