Theorem iuneqfzuz 45617

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 45616	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
3		eqcom 2742	. . . . 5 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
4	3	ralbii 3081	. . . 4 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
5	4	biimpi 216	. . 3 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
6	1	iuneqfzuzlem 45616	. . 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 3950	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   = wceq 1542  ∀wral 3050   ⊆ wss 3900  ∪ ciun 4945  ‘cfv 6491  (class class class)co 7358  ℤ_≥cuz 12753  ...cfz 13425

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-pre-lttri 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-neg 11369  df-z 12491  df-uz 12754  df-fz 13426

This theorem is referenced by:  iundjiun  46741