Theorem dmaddpi 6882

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 4734	. . 3 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2		fnoa 6208	. . . . 5 ⊢ +𝑜 Fn (On × On)
3		fndm 5113	. . . . 5 ⊢ ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
4	2, 3	ax-mp 7	. . . 4 ⊢ dom +𝑜 = (On × On)
5	4	ineq2i 3198	. . 3 ⊢ ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2108	. 2 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 6862	. . 3 ⊢ +N = ( +𝑜 ↾ (N × N))
8	7	dmeqi 4637	. 2 ⊢ dom +N = dom ( +𝑜 ↾ (N × N))
9		df-ni 6861	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3126	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3056	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4427	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3034	. . . . 5 ⊢ N ⊆ On
14		anidm 388	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 144	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4545	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 7	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3013	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 143	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2118	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 102   = wceq 1289   ∖ cdif 2996   ∩ cin 2998   ⊆ wss 2999  ∅c0 3286  {csn 3446  Oncon0 4190  ωcom 4405   × cxp 4436  dom cdm 4438   ↾ cres 4440   Fn wfn 5010   +_𝑜 coa 6178  Ncnpi 6829   +_N cpli 6830

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-ni 6861  df-pli 6862

This theorem is referenced by: (None)