Theorem dmaddpi 10748

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 5946	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8410	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6590	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4157	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2764	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10731	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5847	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10730	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4079	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3966	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7785	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3941	. . . . 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 5636	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3916	. . 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 2774	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 396   = wceq 1540   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4270  {csn 4574   × cxp 5619  dom cdm 5621   ↾ cres 5623  Oncon0 6303  ωcom 7781   +_o coa 8365  Ncnpi 10702   +_N cpli 10703

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373  ax-un 7651

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-fv 6488  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-oadd 8372  df-ni 10730  df-pli 10731

This theorem is referenced by:  addcompi  10752  addasspi  10753  distrpi  10756  addcanpi  10757  addnidpi  10759  ltapi  10761