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Theorem dmaddpi 6629
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
StepHypRef Expression
1 dmres 4680 . . 3 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2 fnoa 6111 . . . . 5 +𝑜 Fn (On × On)
3 fndm 5049 . . . . 5 ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
42, 3ax-mp 7 . . . 4 dom +𝑜 = (On × On)
54ineq2i 3180 . . 3 ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2103 . 2 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 6609 . . 3 +N = ( +𝑜 ↾ (N × N))
87dmeqi 4584 . 2 dom +N = dom ( +𝑜 ↾ (N × N))
9 df-ni 6608 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3108 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3038 . . . . . 6 N ⊆ ω
12 omsson 4381 . . . . . 6 ω ⊆ On
1311, 12sstri 3017 . . . . 5 N ⊆ On
14 anidm 388 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 144 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4493 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 7 . . 3 (N × N) ⊆ (On × On)
18 dfss 2996 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 143 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2113 1 dom +N = (N × N)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  cdif 2979  cin 2981  wss 2982  c0 3267  {csn 3416  Oncon0 4146  ωcom 4359   × cxp 4389  dom cdm 4391  cres 4393   Fn wfn 4947   +𝑜 coa 6082  Ncnpi 6576   +N cpli 6577
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-oadd 6089  df-ni 6608  df-pli 6609
This theorem is referenced by: (None)
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