Theorem dmaddpi 10978

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 6002	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8519	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6643	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4114	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2783	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10961	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5885	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10960	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4072	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3954	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7881	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3917	. . . . 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 5665	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 4095	. . 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 2793	1 ⊢ dom +N = (N × N)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  {csn 4584   × cxp 5648  dom cdm 5650   ↾ cres 5652  Oncon0 6362  ωcom 7877   +_o coa 8476  Ncnpi 10932   +_N cpli 10933

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-oadd 8483  df-ni 10960  df-pli 10961

This theorem is used by:  addcompi  10982  addasspi  10983  distrpi  10986  addcanpi  10987  addnidpi  10989  ltapi  10991