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Theorem iuneqfzuz 40061
 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 40060 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
3 eqcom 2777 . . . . 5 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
43ralbii 3128 . . . 4 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
54biimpi 206 . . 3 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 = 𝑛 ∈ (𝑁...𝑚)𝐴)
61iuneqfzuzlem 40060 . . 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 3767 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 1630  ∀wral 3060   ⊆ wss 3721  ∪ ciun 4652  ‘cfv 6031  (class class class)co 6792  ℤ≥cuz 11887  ...cfz 12532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-pre-lttri 10211 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-neg 10470  df-z 11579  df-uz 11888  df-fz 12533 This theorem is referenced by:  iundjiun  41188
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