Theorem dmmulpi 7327

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 4930	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 6453	. . . . 5 ⊢ ·o Fn (On × On)
3		fndm 5317	. . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom ·o = (On × On)
5	4	ineq2i 3335	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2198	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-mi 7307	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
8	7	dmeqi 4830	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
9		df-ni 7305	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3263	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3189	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4614	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3166	. . . . 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 4735	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3145	. . 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 2208	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 104   = wceq 1353   ∖ cdif 3128   ∩ cin 3130   ⊆ wss 3131  ∅c0 3424  {csn 3594  Oncon0 4365  ωcom 4591   × cxp 4626  dom cdm 4628   ↾ cres 4630   Fn wfn 5213   ·_o comu 6417  Ncnpi 7273   ·_N cmi 7275

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-ni 7305  df-mi 7307

This theorem is referenced by: (None)