Theorem dmmulpi 7421

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

Assertion

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

Proof of Theorem dmmulpi

Step	Hyp	Ref	Expression
1		dmres 4977	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 6526	. . . . 5 ⊢ ·o Fn (On × On)
3		fndm 5367	. . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom ·o = (On × On)
5	4	ineq2i 3370	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2225	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-mi 7401	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
8	7	dmeqi 4877	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
9		df-ni 7399	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3298	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3224	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4659	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3201	. . . . 5 ⊢ N ⊆ On
14		anidm 396	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 146	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4780	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3179	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 145	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2235	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 104   = wceq 1372   ∖ cdif 3162   ∩ cin 3164   ⊆ wss 3165  ∅c0 3459  {csn 3632  Oncon0 4408  ωcom 4636   × cxp 4671  dom cdm 4673   ↾ cres 4675   Fn wfn 5263   ·_o comu 6490  Ncnpi 7367   ·_N cmi 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-oadd 6496  df-omul 6497  df-ni 7399  df-mi 7401

This theorem is referenced by: (None)