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Theorem dmaddpi 7260
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 4902 . . 3 dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2 fnoa 6409 . . . . 5 +o Fn (On × On)
3 fndm 5284 . . . . 5 ( +o Fn (On × On) → dom +o = (On × On))
42, 3ax-mp 5 . . . 4 dom +o = (On × On)
54ineq2i 3318 . . 3 ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
61, 5eqtri 2185 . 2 dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 7240 . . 3 +N = ( +o ↾ (N × N))
87dmeqi 4802 . 2 dom +N = dom ( +o ↾ (N × N))
9 df-ni 7239 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3246 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3172 . . . . . 6 N ⊆ ω
12 omsson 4587 . . . . . 6 ω ⊆ On
1311, 12sstri 3149 . . . . 5 N ⊆ On
14 anidm 394 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 145 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4708 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3128 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 144 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2195 1 dom +N = (N × N)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1342  cdif 3111  cin 3113  wss 3114  c0 3407  {csn 3573  Oncon0 4338  ωcom 4564   × cxp 4599  dom cdm 4601  cres 4603   Fn wfn 5180   +o coa 6375  Ncnpi 7207   +N cpli 7208
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-iinf 4562
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-id 4268  df-iord 4341  df-on 4343  df-suc 4346  df-iom 4565  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-oprab 5843  df-mpo 5844  df-1st 6103  df-2nd 6104  df-recs 6267  df-irdg 6332  df-oadd 6382  df-ni 7239  df-pli 7240
This theorem is referenced by: (None)
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