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Mirrors > Home > MPE Home > Th. List > Mathboxes > resunimafz0 | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2lem3 41404: 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 5464 | . . . . 5 ⊢ (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})) | |
2 | resunimafz0.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) | |
3 | elfzonn0 12380 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(#‘𝐹)) → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
5 | elnn0uz 11601 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
6 | 4, 5 | sylib 207 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
7 | fzisfzounsn 12445 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁})) |
9 | 8 | imaeq2d 5385 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁}))) |
10 | resunimafz0.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼) | |
11 | 10 | ffnd 5959 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(#‘𝐹))) |
12 | fnsnfv 6168 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) | |
13 | 11, 2, 12 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘𝑁)} = (𝐹 “ {𝑁})) |
14 | 13 | uneq2d 3729 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))) |
15 | 1, 9, 14 | 3eqtr4a 2670 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) |
16 | 15 | reseq2d 5317 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}))) |
17 | resundi 5330 | . . 3 ⊢ (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) | |
18 | 16, 17 | syl6eq 2660 | . 2 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)}))) |
19 | resunimafz0.i | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
20 | funfn 5833 | . . . . 5 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
21 | 19, 20 | sylib 207 | . . . 4 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
22 | 10, 2 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼) |
23 | fnressn 6330 | . . . 4 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
24 | 21, 22, 23 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
25 | 24 | uneq2d 3729 | . 2 ⊢ (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
26 | 18, 25 | eqtrd 2644 | 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 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 〈cop 4131 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ℤ≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: trlsegvdeg 41395 |
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