Theorem dmaddpi 7533

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

Assertion

Ref	Expression
dmaddpi	⊢ dom +N = (N × N)

Proof of Theorem dmaddpi

Step	Hyp	Ref	Expression
1		dmres 5030	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 6608	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 5424	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 3403	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2250	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 7513	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 4928	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 7512	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3331	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3257	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4707	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3234	. . . . 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 4829	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3212	. . 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 2260	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 104   = wceq 1395   ∖ cdif 3195   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  {csn 3667  Oncon0 4456  ωcom 4684   × cxp 4719  dom cdm 4721   ↾ cres 4723   Fn wfn 5317   +_o coa 6572  Ncnpi 7480   +_N cpli 7481

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-oadd 6579  df-ni 7512  df-pli 7513

This theorem is referenced by: (None)