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Theorem dmaddpi 9656
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 5378 . . 3 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2 fnoa 7533 . . . . 5 +𝑜 Fn (On × On)
3 fndm 5948 . . . . 5 ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
42, 3ax-mp 5 . . . 4 dom +𝑜 = (On × On)
54ineq2i 3789 . . 3 ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2643 . 2 dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 9639 . . 3 +N = ( +𝑜 ↾ (N × N))
87dmeqi 5285 . 2 dom +N = dom ( +𝑜 ↾ (N × N))
9 df-ni 9638 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3715 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3614 . . . . . 6 N ⊆ ω
12 omsson 7016 . . . . . 6 ω ⊆ On
1311, 12sstri 3592 . . . . 5 N ⊆ On
14 anidm 675 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 221 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 5186 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3570 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 220 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2653 1 dom +N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  cdif 3552  cin 3554  wss 3555  c0 3891  {csn 4148   × cxp 5072  dom cdm 5074  cres 5076  Oncon0 5682   Fn wfn 5842  ωcom 7012   +𝑜 coa 7502  Ncnpi 9610   +N cpli 9611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-oadd 7509  df-ni 9638  df-pli 9639
This theorem is referenced by:  addcompi  9660  addasspi  9661  distrpi  9664  addcanpi  9665  addnidpi  9667  ltapi  9669
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