fargshiftf - Mathbox for Alexander van der Vekens

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Theorem fargshiftf 46098

Description: If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)

**Hypothesis**
Ref	Expression
fargshift.g	⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))

**Assertion**
Ref	Expression
fargshiftf	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)

Distinct variable groups: 𝑥,𝐹 𝑥,𝐸 Allowed substitution hints: 𝐺(𝑥) 𝑁(𝑥)

**Proof of Theorem fargshiftf**
Step	Hyp	Ref	Expression
1		ffn 6717	. . . 4 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
2		fseq1hash 14335	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
3	1, 2	sylan2 593	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
4		eleq1 2821	. . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ ℕ₀ ↔ (♯‘𝐹) ∈ ℕ₀))
5		oveq2 7416	. . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (1...𝑁) = (1...(♯‘𝐹)))
6	5	feq2d 6703	. . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝐹:(1...𝑁)⟶dom 𝐸 ↔ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸))
7	4, 6	anbi12d 631	. . . . 5 ⊢ (𝑁 = (♯‘𝐹) → ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)))
8	7	eqcoms 2740	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)))
9		fz0add1fz1 13701	. . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝑥 + 1) ∈ (1...(♯‘𝐹)))
10		ffvelcdm 7083	. . . . . . . 8 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ (𝑥 + 1) ∈ (1...(♯‘𝐹))) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)
11	10	expcom 414	. . . . . . 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 452	. . . . 5 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸))
14	13	ralrimiv 3145	. . . 4 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸)
15	8, 14	syl6bi 252	. . 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 7109	. 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ↦ cmpt 5231 dom cdm 5676 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 0cc0 11109 1c1 11110 + caddc 11112 ℕ₀cn0 12471 ...cfz 13483 ..^cfzo 13626 ♯chash 14289

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290

This theorem is referenced by: fargshiftf1 46099 fargshiftfo 46100

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