Theorem dmmulpi 10793

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 5968	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8433	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6593	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4166	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2756	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10776	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5850	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10774	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4085	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3977	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7809	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3940	. . . . 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 5636	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3917	. . 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 2766	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1541   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  {csn 4577   × cxp 5619  dom cdm 5621   ↾ cres 5623  Oncon0 6314  ωcom 7805   ·_o comu 8392  Ncnpi 10746   ·_N cmi 10748

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-omul 8399  df-ni 10774  df-mi 10776

This theorem is referenced by:  mulcompi  10798  mulasspi  10799  distrpi  10800  mulcanpi  10802  ltmpi  10806  ordpipq  10844