Theorem dmmulpi 7134

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 4840	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 6346	. . . . 5 ⊢ ·o Fn (On × On)
3		fndm 5222	. . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom ·o = (On × On)
5	4	ineq2i 3274	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2160	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-mi 7114	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
8	7	dmeqi 4740	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
9		df-ni 7112	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3202	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3129	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4526	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3106	. . . . 5 ⊢ N ⊆ On
14		anidm 393	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 145	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4646	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3085	. . 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 2170	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 103   = wceq 1331   ∖ cdif 3068   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  {csn 3527  Oncon0 4285  ωcom 4504   × cxp 4537  dom cdm 4539   ↾ cres 4541   Fn wfn 5118   ·_o comu 6311  Ncnpi 7080   ·_N cmi 7082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-ni 7112  df-mi 7114

This theorem is referenced by: (None)