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Theorem dmaddpi 10504
Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
dmaddpi dom +N = (N × N)

Proof of Theorem dmaddpi
StepHypRef Expression
1 dmres 5873 . . 3 dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2 fnoa 8235 . . . . 5 +o Fn (On × On)
32fndmi 6482 . . . 4 dom +o = (On × On)
43ineq2i 4124 . . 3 ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
51, 4eqtri 2765 . 2 dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6 df-pli 10487 . . 3 +N = ( +o ↾ (N × N))
76dmeqi 5773 . 2 dom +N = dom ( +o ↾ (N × N))
8 df-ni 10486 . . . . . . 7 N = (ω ∖ {∅})
9 difss 4046 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
108, 9eqsstri 3935 . . . . . 6 N ⊆ ω
11 omsson 7648 . . . . . 6 ω ⊆ On
1210, 11sstri 3910 . . . . 5 N ⊆ On
13 anidm 568 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1412, 13mpbir 234 . . . 4 (N ⊆ On ∧ N ⊆ On)
15 xpss12 5566 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1614, 15ax-mp 5 . . 3 (N × N) ⊆ (On × On)
17 dfss 3884 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1816, 17mpbi 233 . 2 (N × N) = ((N × N) ∩ (On × On))
195, 7, 183eqtr4i 2775 1 dom +N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  cdif 3863  cin 3865  wss 3866  c0 4237  {csn 4541   × cxp 5549  dom cdm 5551  cres 5553  Oncon0 6213  ωcom 7644   +o coa 8199  Ncnpi 10458   +N cpli 10459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-oadd 8206  df-ni 10486  df-pli 10487
This theorem is referenced by:  addcompi  10508  addasspi  10509  distrpi  10512  addcanpi  10513  addnidpi  10515  ltapi  10517
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