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Theorem iuneqfzuz 45319
Description: If two unions indexed by upper integers are equal if they agree on any partial indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
iuneqfzuz.z 𝑍 = (ℤ𝑁)
Assertion
Ref Expression
iuneqfzuz (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 = 𝑛𝑍 𝐵)
Distinct variable groups:   𝐴,𝑚   𝐵,𝑚   𝑛,𝑁   𝑚,𝑍,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐵(𝑛)   𝑁(𝑚)

Proof of Theorem iuneqfzuz
StepHypRef Expression
1 iuneqfzuz.z . . 3 𝑍 = (ℤ𝑁)
21iuneqfzuzlem 45318 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
3 eqcom 2743 . . . . 5 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
43ralbii 3092 . . . 4 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
54biimpi 216 . . 3 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
61iuneqfzuzlem 45318 . . 3 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵 𝑛𝑍 𝐴)
75, 6syl 17 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐵 𝑛𝑍 𝐴)
82, 7eqssd 4000 1 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 = 𝑛𝑍 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wral 3060  wss 3950   ciun 4989  cfv 6559  (class class class)co 7429  cuz 12874  ...cfz 13543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208  ax-pre-lttri 11225
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-er 8741  df-en 8982  df-dom 8983  df-sdom 8984  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-neg 11491  df-z 12610  df-uz 12875  df-fz 13544
This theorem is referenced by:  iundjiun  46448
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