fargshiftf - Mathbox for Alexander van der Vekens

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Theorem fargshiftf 46843

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 14367	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
3	1, 2	sylan2 591	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
4		eleq1 2813	. . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ ℕ₀ ↔ (♯‘𝐹) ∈ ℕ₀))
5		oveq2 7424	. . . . . . 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 630	. . . . 5 ⊢ (𝑁 = (♯‘𝐹) → ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)))
8	7	eqcoms 2733	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸)))
9		fz0add1fz1 13734	. . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑥 ∈ (0..^(♯‘𝐹))) → (𝑥 + 1) ∈ (1...(♯‘𝐹)))
10		ffvelcdm 7086	. . . . . . . 8 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ (𝑥 + 1) ∈ (1...(♯‘𝐹))) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸)
11	10	expcom 412	. . . . . . 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 450	. . . . 5 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝐹‘(𝑥 + 1)) ∈ dom 𝐸))
14	13	ralrimiv 3135	. . . 4 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → ∀𝑥 ∈ (0..^(♯‘𝐹))(𝐹‘(𝑥 + 1)) ∈ dom 𝐸)
15	8, 14	biimtrdi 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 7115	. 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ↦ cmpt 5226 dom cdm 5672 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 ℕ₀cn0 12502 ...cfz 13516 ..^cfzo 13659 ♯chash 14321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322

This theorem is referenced by: fargshiftf1 46844 fargshiftfo 46845

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