Theorem dmmulpi 10843

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 5994	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8472	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6620	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4167	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2784	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10826	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5876	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10824	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4087	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3980	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7845	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3943	. . . . 5 ⊢ N ⊆ On
13		anidm 572	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 233	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5658	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3921	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 232	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2794	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 399   = wceq 1559   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4579   × cxp 5641  dom cdm 5643   ↾ cres 5645  Oncon0 6341  ωcom 7841   ·_o comu 8429  Ncnpi 10796   ·_N cmi 10798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-omul 8436  df-ni 10824  df-mi 10826

This theorem is referenced by:  mulcompi  10848  mulasspi  10849  distrpi  10850  mulcanpi  10852  ltmpi  10856  ordpipq  10894