Theorem dmaddpi 10956

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 6003	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8500	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6635	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4163	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2784	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10939	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5886	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10938	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4083	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3977	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7870	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3940	. . . . 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 5666	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3918	. . 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 2794	1 ⊢ dom +N = (N × N)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584   × cxp 5649  dom cdm 5651   ↾ cres 5653  Oncon0 6355  ωcom 7866   +_o coa 8457  Ncnpi 10910   +_N cpli 10911

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-oadd 8464  df-ni 10938  df-pli 10939

This theorem is used by:  addcompi  10960  addasspi  10961  distrpi  10964  addcanpi  10965  addnidpi  10967  ltapi  10969