Theorem dmaddpi 7126

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 4835	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 6336	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 5217	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 3269	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2158	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 7106	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 4735	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 7105	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3197	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3124	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4521	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3101	. . . . 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 4641	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3080	. . 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 2168	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 103   = wceq 1331   ∖ cdif 3063   ∩ cin 3065   ⊆ wss 3066  ∅c0 3358  {csn 3522  Oncon0 4280  ωcom 4499   × cxp 4532  dom cdm 4534   ↾ cres 4536   Fn wfn 5113   +_o coa 6303  Ncnpi 7073   +_N cpli 7074

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-ni 7105  df-pli 7106

This theorem is referenced by: (None)