Theorem dmaddpi 9712

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 5419	. . 3 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2		fnoa 7588	. . . . 5 ⊢ +𝑜 Fn (On × On)
3		fndm 5990	. . . . 5 ⊢ ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +𝑜 = (On × On)
5	4	ineq2i 3811	. . 3 ⊢ ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2644	. 2 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 9695	. . 3 ⊢ +N = ( +𝑜 ↾ (N × N))
8	7	dmeqi 5325	. 2 ⊢ dom +N = dom ( +𝑜 ↾ (N × N))
9		df-ni 9694	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3737	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3635	. . . . . 6 ⊢ N ⊆ ω
12		omsson 7069	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3612	. . . . 5 ⊢ N ⊆ On
14		anidm 676	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 221	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 5225	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3589	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 220	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2654	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 384   = wceq 1483   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   × cxp 5112  dom cdm 5114   ↾ cres 5116  Oncon0 5723   Fn wfn 5883  ωcom 7065   +_𝑜 coa 7557  Ncnpi 9666   +_N cpli 9667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-oadd 7564  df-ni 9694  df-pli 9695

This theorem is referenced by:  addcompi  9716  addasspi  9717  distrpi  9720  addcanpi  9721  addnidpi  9723  ltapi  9725