Theorem dmaddpi 10845

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 5996	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8472	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6621	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4169	. . 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 10828	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5878	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10827	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4089	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3982	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7846	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3945	. . . . 5 ⊢ N ⊆ On
13		anidm 572	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 233	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5660	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3923	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 232	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2794	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 399   = wceq 1559   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4581   × cxp 5643  dom cdm 5645   ↾ cres 5647  Oncon0 6342  ωcom 7842   +_o coa 8429  Ncnpi 10799   +_N cpli 10800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-oadd 8436  df-ni 10827  df-pli 10828

This theorem is referenced by:  addcompi  10849  addasspi  10850  distrpi  10853  addcanpi  10854  addnidpi  10856  ltapi  10858