Theorem dmmulpi 7076

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 4796	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 6298	. . . . 5 ⊢ ·o Fn (On × On)
3		fndm 5178	. . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
4	2, 3	ax-mp 7	. . . 4 ⊢ dom ·o = (On × On)
5	4	ineq2i 3238	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2133	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-mi 7056	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
8	7	dmeqi 4698	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
9		df-ni 7054	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3166	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3093	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4484	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3070	. . . . 5 ⊢ N ⊆ On
14		anidm 391	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 145	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4604	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 7	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3049	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 144	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2143	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 103   = wceq 1312   ∖ cdif 3032   ∩ cin 3034   ⊆ wss 3035  ∅c0 3327  {csn 3491  Oncon0 4243  ωcom 4462   × cxp 4495  dom cdm 4497   ↾ cres 4499   Fn wfn 5074   ·_o comu 6263  Ncnpi 7022   ·_N cmi 7024

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-oadd 6269  df-omul 6270  df-ni 7054  df-mi 7056

This theorem is referenced by: (None)