Theorem iuneqfzuz 45439

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 45438	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
3		eqcom 2738	. . . . 5 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
4	3	ralbii 3078	. . . 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 45438	. . 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 3947	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3047   ⊆ wss 3897  ∪ ciun 4941  ‘cfv 6487  (class class class)co 7352  ℤ_≥cuz 12738  ...cfz 13413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-pre-lttri 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-neg 11353  df-z 12475  df-uz 12739  df-fz 13414

This theorem is referenced by:  iundjiun  46563