Theorem dmaddpi 10893

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 6016	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8502	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6646	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4173	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2789	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10876	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5899	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10875	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4093	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3986	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7875	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3949	. . . . 5 ⊢ N ⊆ On
13		anidm 575	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5681	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3927	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 233	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2799	1 ⊢ dom +N = (N × N)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4594   × cxp 5664  dom cdm 5666   ↾ cres 5668  Oncon0 6367  ωcom 7871   +_o coa 8459  Ncnpi 10847   +_N cpli 10848

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-oadd 8466  df-ni 10875  df-pli 10876

This theorem is used by:  addcompi  10897  addasspi  10898  distrpi  10901  addcanpi  10902  addnidpi  10904  ltapi  10906