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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iundisjcnt | Structured version Visualization version GIF version |
Description: Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
Ref | Expression |
---|---|
iundisjcnt.0 | ⊢ Ⅎ𝑛𝐵 |
iundisjcnt.1 | ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) |
iundisjcnt.2 | ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀))) |
Ref | Expression |
---|---|
iundisjcnt | ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2939 | . . . 4 ⊢ Ⅎ𝑘𝐴 | |
2 | iundisjcnt.0 | . . . 4 ⊢ Ⅎ𝑛𝐵 | |
3 | iundisjcnt.1 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) | |
4 | 1, 2, 3 | iundisjf 29911 | . . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) |
5 | simpr 478 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = ℕ) → 𝑁 = ℕ) | |
6 | 5 | iuneq1d 4733 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = ℕ) → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ ℕ 𝐴) |
7 | 5 | iuneq1d 4733 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = ℕ) → ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
8 | 4, 6, 7 | 3eqtr4a 2857 | . 2 ⊢ ((𝜑 ∧ 𝑁 = ℕ) → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
9 | 2, 3 | iundisjfi 30065 | . . 3 ⊢ ∪ 𝑛 ∈ (1..^𝑀)𝐴 = ∪ 𝑛 ∈ (1..^𝑀)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) |
10 | simpr 478 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = (1..^𝑀)) → 𝑁 = (1..^𝑀)) | |
11 | 10 | iuneq1d 4733 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = (1..^𝑀)) → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ (1..^𝑀)𝐴) |
12 | 10 | iuneq1d 4733 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = (1..^𝑀)) → ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ∪ 𝑛 ∈ (1..^𝑀)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
13 | 9, 11, 12 | 3eqtr4a 2857 | . 2 ⊢ ((𝜑 ∧ 𝑁 = (1..^𝑀)) → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
14 | iundisjcnt.2 | . 2 ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀))) | |
15 | 8, 13, 14 | mpjaodan 982 | 1 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 Ⅎwnfc 2926 ∖ cdif 3764 ∪ ciun 4708 (class class class)co 6876 1c1 10223 ℕcn 11310 ..^cfzo 12716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-fzo 12717 |
This theorem is referenced by: measiuns 30788 |
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