Theorem dmaddpi 10863

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 8481	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6629	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4172	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2788	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10846	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5885	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10845	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4092	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3985	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7854	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3948	. . . . 5 ⊢ N ⊆ On
13		anidm 574	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5667	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3926	. . 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 2798	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 400   = wceq 1563   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  {csn 4585   × cxp 5650  dom cdm 5652   ↾ cres 5654  Oncon0 6350  ωcom 7850   +_o coa 8438  Ncnpi 10817   +_N cpli 10818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-oadd 8445  df-ni 10845  df-pli 10846

This theorem is referenced by:  addcompi  10867  addasspi  10868  distrpi  10871  addcanpi  10872  addnidpi  10874  ltapi  10876