Theorem dmaddpi 10930

Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
dmaddpi	⊢ dom +N = (N × N)

Proof of Theorem dmaddpi

Step	Hyp	Ref	Expression
1		dmres 6030	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8546	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6672	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4217	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2765	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10913	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5915	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10912	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4136	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4030	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7891	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3993	. . . . 5 ⊢ N ⊆ On
13		anidm 564	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 231	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5700	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3970	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 230	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2775	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1540   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626   × cxp 5683  dom cdm 5685   ↾ cres 5687  Oncon0 6384  ωcom 7887   +_o coa 8503  Ncnpi 10884   +_N cpli 10885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-oadd 8510  df-ni 10912  df-pli 10913

This theorem is referenced by:  addcompi  10934  addasspi  10935  distrpi  10938  addcanpi  10939  addnidpi  10941  ltapi  10943