Theorem dmaddpi 10314

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

Step	Hyp	Ref	Expression
1		dmres 5877	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8135	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 6457	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 4188	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2846	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 10297	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 5775	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 10296	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 4110	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 4003	. . . . . 6 ⊢ N ⊆ ω
12		omsson 7586	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3978	. . . . 5 ⊢ N ⊆ On
14		anidm 567	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 233	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 5572	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3955	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 232	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2856	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 398   = wceq 1537   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  {csn 4569   × cxp 5555  dom cdm 5557   ↾ cres 5559  Oncon0 6193   Fn wfn 6352  ωcom 7582   +_o coa 8101  Ncnpi 10268   +_N cpli 10269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-oadd 8108  df-ni 10296  df-pli 10297

This theorem is referenced by:  addcompi  10318  addasspi  10319  distrpi  10322  addcanpi  10323  addnidpi  10325  ltapi  10327