Theorem dmaddpi 10921

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 6021	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8535	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6663	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4211	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2756	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10904	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5911	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10903	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4132	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4016	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7880	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3991	. . . . 5 ⊢ N ⊆ On
13		anidm 563	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5697	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3967	. . 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 2766	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 394   = wceq 1533   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  {csn 4632   × cxp 5680  dom cdm 5682   ↾ cres 5684  Oncon0 6374  ωcom 7876   +_o coa 8490  Ncnpi 10875   +_N cpli 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-oadd 8497  df-ni 10903  df-pli 10904

This theorem is referenced by:  addcompi  10925  addasspi  10926  distrpi  10929  addcanpi  10930  addnidpi  10932  ltapi  10934