Theorem iuneqfzuz 45239

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 45238	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
3		eqcom 2747	. . . . 5 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ↔ ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
4	3	ralbii 3099	. . . 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 45238	. . 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 4026	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   = wceq 1537  ∀wral 3067   ⊆ wss 3976  ∪ ciun 5015  ‘cfv 6568  (class class class)co 7443  ℤ_≥cuz 12897  ...cfz 13561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764  ax-cnex 11234  ax-resscn 11235  ax-pre-lttri 11252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8024  df-2nd 8025  df-er 8757  df-en 8998  df-dom 8999  df-sdom 9000  df-pnf 11320  df-mnf 11321  df-xr 11322  df-ltxr 11323  df-le 11324  df-neg 11517  df-z 12634  df-uz 12898  df-fz 13562

This theorem is referenced by:  iundjiun  46370