Theorem dmaddpi 10903

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 6009	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8499	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6640	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4166	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2785	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10886	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5892	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10885	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4086	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3980	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7870	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3943	. . . . 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 5674	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3921	. . 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 2795	1 ⊢ dom +N = (N × N)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587   × cxp 5657  dom cdm 5659   ↾ cres 5661  Oncon0 6361  ωcom 7866   +_o coa 8456  Ncnpi 10857   +_N cpli 10858

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-oadd 8463  df-ni 10885  df-pli 10886

This theorem is used by:  addcompi  10907  addasspi  10908  distrpi  10911  addcanpi  10912  addnidpi  10914  ltapi  10916