Theorem dmaddpi 10504

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 5873	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8235	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6482	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4124	. . 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 10487	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5773	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10486	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4046	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3935	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7648	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3910	. . . . 5 ⊢ N ⊆ On
13		anidm 568	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5566	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3884	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 233	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2775	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 399   = wceq 1543   ∖ cdif 3863   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  {csn 4541   × cxp 5549  dom cdm 5551   ↾ cres 5553  Oncon0 6213  ωcom 7644   +_o coa 8199  Ncnpi 10458   +_N cpli 10459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-oadd 8206  df-ni 10486  df-pli 10487

This theorem is referenced by:  addcompi  10508  addasspi  10509  distrpi  10512  addcanpi  10513  addnidpi  10515  ltapi  10517