Theorem dmmulpi 10858

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 5986	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8482	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6633	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4196	. . 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 10841	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5887	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10839	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4118	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4003	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7833	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3978	. . . . 5 ⊢ N ⊆ On
13		anidm 565	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5675	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3953	. . 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 396   = wceq 1541   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  ∅c0 4309  {csn 4613   × cxp 5658  dom cdm 5660   ↾ cres 5662  Oncon0 6344  ωcom 7829   ·_o comu 8437  Ncnpi 10811   ·_N cmi 10813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pr 5411  ax-un 7699

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-oprab 7388  df-mpo 7389  df-om 7830  df-1st 7948  df-2nd 7949  df-omul 8444  df-ni 10839  df-mi 10841

This theorem is referenced by:  mulcompi  10863  mulasspi  10864  distrpi  10865  mulcanpi  10867  ltmpi  10871  ordpipq  10909