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Mirrors > Home > MPE Home > Th. List > resunimafz0 | Structured version Visualization version GIF version |
Description: TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 28021: 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 5975 | . . . . 5 ⊢ (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})) | |
2 | resunimafz0.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
3 | elfzonn0 13077 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
5 | elnn0uz 12271 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
6 | 4, 5 | sylib 221 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
7 | fzisfzounsn 13144 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) |
9 | 8 | imaeq2d 5896 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁}))) |
10 | resunimafz0.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
11 | 10 | ffnd 6488 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹))) |
12 | fnsnfv 6718 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) | |
13 | 11, 2, 12 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) |
14 | 13 | uneq2d 4090 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))) |
15 | 1, 9, 14 | 3eqtr4a 2859 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) |
16 | 15 | reseq2d 5818 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}))) |
17 | resundi 5832 | . . 3 ⊢ (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) | |
18 | 16, 17 | eqtrdi 2849 | . 2 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)}))) |
19 | resunimafz0.i | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
20 | 19 | funfnd 6355 | . . . 4 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
21 | 10, 2 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼) |
22 | fnressn 6897 | . . . 4 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
23 | 20, 21, 22 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
24 | 23 | uneq2d 4090 | . 2 ⊢ (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
25 | 18, 24 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 {csn 4525 〈cop 4531 dom cdm 5519 ↾ cres 5521 “ cima 5522 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℕ0cn0 11885 ℤ≥cuz 12231 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: trlsegvdeg 28012 |
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