Theorem iuneqfzuz 42764

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 = (ℤ≥‘𝑁)

Assertion

Ref	Expression
iuneqfzuz	⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 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

Step	Hyp	Ref	Expression
1		iuneqfzuz.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑁)
2	1	iuneqfzuzlem 42763	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
3		eqcom 2745	. . . . 5 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
4	3	ralbii 3090	. . . 4 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
5	4	biimpi 215	. . 3 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
6	1	iuneqfzuzlem 42763	. . 3 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 → ∪ 𝑛 ∈ 𝑍 𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐴)
7	5, 6	syl 17	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐴)
8	2, 7	eqssd 3934	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   = wceq 1539  ∀wral 3063   ⊆ wss 3883  ∪ ciun 4921  ‘cfv 6418  (class class class)co 7255  ℤ_≥cuz 12511  ...cfz 13168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-neg 11138  df-z 12250  df-uz 12512  df-fz 13169

This theorem is referenced by:  iundjiun  43888