Theorem dmaddpi 10813

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 5979	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8445	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6604	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4171	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2760	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10796	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5861	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10795	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4090	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3982	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7822	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3945	. . . . 5 ⊢ N ⊆ On
13		anidm 564	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 231	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5647	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3922	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 230	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2770	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1542   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582   × cxp 5630  dom cdm 5632   ↾ cres 5634  Oncon0 6325  ωcom 7818   +_o coa 8404  Ncnpi 10767   +_N cpli 10768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-oadd 8411  df-ni 10795  df-pli 10796

This theorem is referenced by:  addcompi  10817  addasspi  10818  distrpi  10821  addcanpi  10822  addnidpi  10824  ltapi  10826