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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
StepHypRef 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))
42, 3ax-mp 7 . . . 4 dom +𝑜 = (On × On)
54ineq2i 3198 . . 3 ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2108 . 2 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 6862 . . 3 +N = ( +𝑜 ↾ (N × N))
87dmeqi 4637 . 2 dom +N = dom ( +𝑜 ↾ (N × N))
9 df-ni 6861 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3126 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3056 . . . . . 6 N ⊆ ω
12 omsson 4427 . . . . . 6 ω ⊆ On
1311, 12sstri 3034 . . . . 5 N ⊆ On
14 anidm 388 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 144 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4545 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 7 . . 3 (N × N) ⊆ (On × On)
18 dfss 3013 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 143 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2118 1 dom +N = (N × N)
Colors of variables: wff set class
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)
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