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Theorem dmaddpi 7140
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 4840 . . 3 dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2 fnoa 6343 . . . . 5 +o Fn (On × On)
3 fndm 5222 . . . . 5 ( +o Fn (On × On) → dom +o = (On × On))
42, 3ax-mp 5 . . . 4 dom +o = (On × On)
54ineq2i 3274 . . 3 ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
61, 5eqtri 2160 . 2 dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 7120 . . 3 +N = ( +o ↾ (N × N))
87dmeqi 4740 . 2 dom +N = dom ( +o ↾ (N × N))
9 df-ni 7119 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3202 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3129 . . . . . 6 N ⊆ ω
12 omsson 4526 . . . . . 6 ω ⊆ On
1311, 12sstri 3106 . . . . 5 N ⊆ On
14 anidm 393 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 145 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4646 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3085 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 144 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2170 1 dom +N = (N × N)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  cdif 3068  cin 3070  wss 3071  c0 3363  {csn 3527  Oncon0 4285  ωcom 4504   × cxp 4537  dom cdm 4539  cres 4541   Fn wfn 5118   +o coa 6310  Ncnpi 7087   +N cpli 7088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-ni 7119  df-pli 7120
This theorem is referenced by: (None)
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