Theorem dmmulpi 10889

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

Assertion

Ref	Expression
dmmulpi	⊢ dom ·N = (N × N)

Proof of Theorem dmmulpi

Step	Hyp	Ref	Expression
1		dmres 6003	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8512	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6653	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4209	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2759	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10872	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5904	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10870	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4131	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4016	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7862	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3991	. . . . 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 5691	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3966	. . 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 2769	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1540   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  {csn 4628   × cxp 5674  dom cdm 5676   ↾ cres 5678  Oncon0 6364  ωcom 7858   ·_o comu 8467  Ncnpi 10842   ·_N cmi 10844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-omul 8474  df-ni 10870  df-mi 10872

This theorem is referenced by:  mulcompi  10894  mulasspi  10895  distrpi  10896  mulcanpi  10898  ltmpi  10902  ordpipq  10940