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Theorem iuneqfzuz 43723
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 43722 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
3 eqcom 2738 . . . . 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 215 . . 3 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
61iuneqfzuzlem 43722 . . 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 3979 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 1541  wral 3060  wss 3928   ciun 4974  cfv 6516  (class class class)co 7377  cuz 12787  ...cfz 13449
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-neg 11412  df-z 12524  df-uz 12788  df-fz 13450
This theorem is referenced by:  iundjiun  44854
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