Theorem dmaddpi 10881

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 6001	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8504	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6650	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4208	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2760	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10864	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5902	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10863	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4130	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4015	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7855	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3990	. . . . 5 ⊢ N ⊆ On
13		anidm 565	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5690	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3965	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 229	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2770	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 396   = wceq 1541   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627   × cxp 5673  dom cdm 5675   ↾ cres 5677  Oncon0 6361  ωcom 7851   +_o coa 8459  Ncnpi 10835   +_N cpli 10836

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-oadd 8466  df-ni 10863  df-pli 10864

This theorem is referenced by:  addcompi  10885  addasspi  10886  distrpi  10889  addcanpi  10890  addnidpi  10892  ltapi  10894