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Theorem iuneqfzuz 45235
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 45234 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
3 eqcom 2740 . . . . 5 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
43ralbii 3089 . . . 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 45234 . . 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 4013 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 1535  wral 3057  wss 3963   ciun 4998  cfv 6558  (class class class)co 7425  cuz 12869  ...cfz 13537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747  ax-cnex 11202  ax-resscn 11203  ax-pre-lttri 11220
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  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 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-er 8738  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-neg 11486  df-z 12605  df-uz 12870  df-fz 13538
This theorem is referenced by:  iundjiun  46366
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