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Mirrors > Home > ILE Home > Th. List > resunimafz0 | GIF version |
Description: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.) |
Ref | Expression |
---|---|
resunimafz0.i | ⊢ (𝜑 → Fun 𝐼) |
resunimafz0.f | ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) |
resunimafz0.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
Ref | Expression |
---|---|
resunimafz0 | ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaundi 5010 | . . . . 5 ⊢ (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})) | |
2 | resunimafz0.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
3 | elfzonn0 10111 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
5 | elnn0uz 9494 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
6 | 4, 5 | sylib 121 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
7 | fzisfzounsn 10161 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) | |
8 | 6, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) |
9 | 8 | imaeq2d 4940 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁}))) |
10 | resunimafz0.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
11 | 10 | ffnd 5332 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹))) |
12 | fnsnfv 5539 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) | |
13 | 11, 2, 12 | syl2anc 409 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) |
14 | 13 | uneq2d 3271 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))) |
15 | 1, 9, 14 | 3eqtr4a 2223 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) |
16 | 15 | reseq2d 4878 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}))) |
17 | resundi 4891 | . . 3 ⊢ (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) | |
18 | 16, 17 | eqtrdi 2213 | . 2 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)}))) |
19 | resunimafz0.i | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
20 | funfn 5212 | . . . . 5 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
21 | 19, 20 | sylib 121 | . . . 4 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
22 | 10, 2 | ffvelrnd 5615 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼) |
23 | fnressn 5665 | . . . 4 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
24 | 21, 22, 23 | syl2anc 409 | . . 3 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
25 | 24 | uneq2d 3271 | . 2 ⊢ (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
26 | 18, 25 | eqtrd 2197 | 1 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 {csn 3570 〈cop 3573 dom cdm 4598 ↾ cres 4600 “ cima 4601 Fun wfun 5176 Fn wfn 5177 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 0cc0 7744 ℕ0cn0 9105 ℤ≥cuz 9457 ...cfz 9935 ..^cfzo 10067 ♯chash 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-fzo 10068 |
This theorem is referenced by: (None) |
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