Theorem dmaddpi

Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	dmaddpi	⊢ dom +N = ( N × N )

Proof

Step	Hyp	Ref	Expression
1		dmres	⊢ dom ( +o ↾ ( N × N ) ) = ( ( N × N ) ∩ dom +o )
2		fnoa	⊢ +o Fn ( On × On )
3	2	fndmi	⊢ dom +o = ( On × On )
4	3	ineq2i	⊢ ( ( N × N ) ∩ dom +o ) = ( ( N × N ) ∩ ( On × On ) )
5	1 4	eqtri	⊢ dom ( +o ↾ ( N × N ) ) = ( ( N × N ) ∩ ( On × On ) )
6		df-pli	⊢ +N = ( +o ↾ ( N × N ) )
7	6	dmeqi	⊢ dom +N = dom ( +o ↾ ( N × N ) )
8		df-ni	⊢ N = ( ω ∖ { ∅ } )
9		difss	⊢ ( ω ∖ { ∅ } ) ⊆ ω
10	8 9	eqsstri	⊢ N ⊆ ω
11		omsson	⊢ ω ⊆ On
12	10 11	sstri	⊢ N ⊆ On
13		anidm	⊢ ( ( N ⊆ On ∧ N ⊆ On ) ↔ N ⊆ On )
14	12 13	mpbir	⊢ ( N ⊆ On ∧ N ⊆ On )
15		xpss12	⊢ ( ( N ⊆ On ∧ N ⊆ On ) → ( N × N ) ⊆ ( On × On ) )
16	14 15	ax-mp	⊢ ( N × N ) ⊆ ( On × On )
17		dfss	⊢ ( ( N × N ) ⊆ ( On × On ) ↔ ( N × N ) = ( ( N × N ) ∩ ( On × On ) ) )
18	16 17	mpbi	⊢ ( N × N ) = ( ( N × N ) ∩ ( On × On ) )
19	5 7 18	3eqtr4i	⊢ dom +N = ( N × N )