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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 5037 | . . . . 5 ⊢ (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})) | |
2 | resunimafz0.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
3 | elfzonn0 10172 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
5 | elnn0uz 9554 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
6 | 4, 5 | sylib 122 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
7 | fzisfzounsn 10222 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) | |
8 | 6, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) |
9 | 8 | imaeq2d 4966 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁}))) |
10 | resunimafz0.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
11 | 10 | ffnd 5362 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹))) |
12 | fnsnfv 5571 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) | |
13 | 11, 2, 12 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) |
14 | 13 | uneq2d 3289 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))) |
15 | 1, 9, 14 | 3eqtr4a 2236 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) |
16 | 15 | reseq2d 4903 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}))) |
17 | resundi 4916 | . . 3 ⊢ (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) | |
18 | 16, 17 | eqtrdi 2226 | . 2 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)}))) |
19 | resunimafz0.i | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
20 | funfn 5242 | . . . . 5 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
21 | 19, 20 | sylib 122 | . . . 4 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
22 | 10, 2 | ffvelcdmd 5648 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼) |
23 | fnressn 5698 | . . . 4 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
24 | 21, 22, 23 | syl2anc 411 | . . 3 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
25 | 24 | uneq2d 3289 | . 2 ⊢ (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
26 | 18, 25 | eqtrd 2210 | 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 1353 ∈ wcel 2148 ∪ cun 3127 {csn 3591 〈cop 3594 dom cdm 4623 ↾ cres 4625 “ cima 4626 Fun wfun 5206 Fn wfn 5207 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 0cc0 7802 ℕ0cn0 9165 ℤ≥cuz 9517 ...cfz 9995 ..^cfzo 10128 ♯chash 10739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-fz 9996 df-fzo 10129 |
This theorem is referenced by: (None) |
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