Theorem dmmulpi 10748

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 5945	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8410	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6589	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4156	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2764	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10731	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5846	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10729	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4078	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3966	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7784	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3941	. . . . 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 5635	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3916	. . 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 2774	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 396   = wceq 1540   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4269  {csn 4573   × cxp 5618  dom cdm 5620   ↾ cres 5622  Oncon0 6302  ωcom 7780   ·_o comu 8365  Ncnpi 10701   ·_N cmi 10703

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-omul 8372  df-ni 10729  df-mi 10731

This theorem is referenced by:  mulcompi  10753  mulasspi  10754  distrpi  10755  mulcanpi  10757  ltmpi  10761  ordpipq  10799