Theorem dmaddpi 10884

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 5996	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8506	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6646	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4204	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2754	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10867	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5897	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10866	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4126	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4011	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7855	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3986	. . . . 5 ⊢ N ⊆ On
13		anidm 564	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5684	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3961	. . 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 2764	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1533   ∖ cdif 3940   ∩ cin 3942   ⊆ wss 3943  ∅c0 4317  {csn 4623   × cxp 5667  dom cdm 5669   ↾ cres 5671  Oncon0 6357  ωcom 7851   +_o coa 8461  Ncnpi 10838   +_N cpli 10839

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-oadd 8468  df-ni 10866  df-pli 10867

This theorem is referenced by:  addcompi  10888  addasspi  10889  distrpi  10892  addcanpi  10893  addnidpi  10895  ltapi  10897