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 6514 | . . . 4 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁)) | |
2 | fseq1hash 13738 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁) | |
3 | 1, 2 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁) |
4 | eleq1 2900 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)) | |
5 | oveq2 7164 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (1...𝑁) = (1...(♯‘𝐹))) | |
6 | 5 | feq2d 6500 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝐹:(1...𝑁)⟶dom 𝐸 ↔ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)) |
7 | 4, 6 | anbi12d 632 | . . . . 5 ⊢ (𝑁 = (♯‘𝐹) → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))) |
8 | 7 | eqcoms 2829 | . . . 4 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))) |
9 | fz0add1fz1 13108 | . . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝑥 + 1) ∈ (1...(♯‘𝐹))) | |
10 | ffvelrn 6849 | . . . . . . . 8 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ (𝑥 + 1) ∈ (1...(♯‘𝐹))) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸) | |
11 | 10 | expcom 416 | . . . . . . 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 454 | . . . . 5 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
14 | 13 | ralrimiv 3181 | . . . 4 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸) |
15 | 8, 14 | syl6bi 255 | . . 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 6874 | . 2 ⊢ (∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸 ↔ 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
19 | 16, 18 | sylib 220 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ↦ cmpt 5146 dom cdm 5555 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 ℕ0cn0 11898 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 |
This theorem is referenced by: fargshiftf1 43621 fargshiftfo 43622 |
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