Theorem dmaddpi 10811

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 5971	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8440	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6596	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4153	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2763	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10794	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5853	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10793	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4073	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3968	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7817	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3931	. . . . 5 ⊢ N ⊆ On
13		anidm 569	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 232	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5640	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3909	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 231	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2773	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 396   = wceq 1547   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562   × cxp 5623  dom cdm 5625   ↾ cres 5627  Oncon0 6317  ωcom 7813   +_o coa 8399  Ncnpi 10765   +_N cpli 10766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-oadd 8406  df-ni 10793  df-pli 10794

This theorem is referenced by:  addcompi  10815  addasspi  10816  distrpi  10819  addcanpi  10820  addnidpi  10822  ltapi  10824