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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 6672 | . . . 4 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁)) | |
2 | fseq1hash 14285 | . . . 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 2822 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)) | |
5 | oveq2 7369 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (1...𝑁) = (1...(♯‘𝐹))) | |
6 | 5 | feq2d 6658 | . . . . . 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 2741 | . . . 4 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))) |
9 | fz0add1fz1 13651 | . . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝑥 + 1) ∈ (1...(♯‘𝐹))) | |
10 | ffvelcdm 7036 | . . . . . . . 8 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ (𝑥 + 1) ∈ (1...(♯‘𝐹))) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸) | |
11 | 10 | expcom 415 | . . . . . . 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 453 | . . . . 5 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)) |
14 | 13 | ralrimiv 3139 | . . . 4 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸) |
15 | 8, 14 | syl6bi 253 | . . 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 7062 | . 2 ⊢ (∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸 ↔ 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
19 | 16, 18 | sylib 217 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ↦ cmpt 5192 dom cdm 5637 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 + caddc 11062 ℕ0cn0 12421 ...cfz 13433 ..^cfzo 13576 ♯chash 14239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 |
This theorem is referenced by: fargshiftf1 45723 fargshiftfo 45724 |
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