Theorem dmmulpi 10785

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 5963	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8427	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6586	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4168	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2752	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10768	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5847	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10766	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4087	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3982	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7803	. . . . . 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 5634	. . . 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 2762	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1540   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {csn 4577   × cxp 5617  dom cdm 5619   ↾ cres 5621  Oncon0 6307  ωcom 7799   ·_o comu 8386  Ncnpi 10738   ·_N cmi 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-omul 8393  df-ni 10766  df-mi 10768

This theorem is referenced by:  mulcompi  10790  mulasspi  10791  distrpi  10792  mulcanpi  10794  ltmpi  10798  ordpipq  10836