Theorem dmaddpi 6515

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 4650	. . 3 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2		fnoa 6050	. . . . 5 ⊢ +𝑜 Fn (On × On)
3		fndm 5018	. . . . 5 ⊢ ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
4	2, 3	ax-mp 7	. . . 4 ⊢ dom +𝑜 = (On × On)
5	4	ineq2i 3164	. . 3 ⊢ ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2101	. 2 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 6495	. . 3 ⊢ +N = ( +𝑜 ↾ (N × N))
8	7	dmeqi 4554	. 2 ⊢ dom +N = dom ( +𝑜 ↾ (N × N))
9		df-ni 6494	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3098	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3029	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4353	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3008	. . . . 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 4463	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 7	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 2987	. . 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 2111	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 102   = wceq 1284   ∖ cdif 2970   ∩ cin 2972   ⊆ wss 2973  ∅c0 3251  {csn 3398  Oncon0 4118  ωcom 4331   × cxp 4361  dom cdm 4363   ↾ cres 4365   Fn wfn 4917   +_𝑜 coa 6021  Ncnpi 6462   +_N cpli 6463

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-ni 6494  df-pli 6495

This theorem is referenced by: (None)