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Theorem dmaddpi 9750
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 5454 . . 3 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2 fnoa 7633 . . . . 5 +𝑜 Fn (On × On)
3 fndm 6028 . . . . 5 ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
42, 3ax-mp 5 . . . 4 dom +𝑜 = (On × On)
54ineq2i 3844 . . 3 ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2673 . 2 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 9733 . . 3 +N = ( +𝑜 ↾ (N × N))
87dmeqi 5357 . 2 dom +N = dom ( +𝑜 ↾ (N × N))
9 df-ni 9732 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3770 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3668 . . . . . 6 N ⊆ ω
12 omsson 7111 . . . . . 6 ω ⊆ On
1311, 12sstri 3645 . . . . 5 N ⊆ On
14 anidm 677 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 221 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 5158 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3622 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 220 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2683 1 dom +N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  cres 5145  Oncon0 5761   Fn wfn 5921  ωcom 7107   +𝑜 coa 7602  Ncnpi 9704   +N cpli 9705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-oadd 7609  df-ni 9732  df-pli 9733
This theorem is referenced by:  addcompi  9754  addasspi  9755  distrpi  9758  addcanpi  9759  addnidpi  9761  ltapi  9763
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