Theorem dmaddpi 7686

Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)

Assertion

Ref	Expression
dmaddpi	⊢ dom +N = (N × N)

Proof of Theorem dmaddpi

Step	Hyp	Ref	Expression
1		dmres 5082	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 6714	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 5478	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 3429	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2259	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 7666	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 4980	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 7665	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3355	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3280	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4758	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3257	. . . . 5 ⊢ N ⊆ On
14		anidm 400	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 146	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4880	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3234	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 145	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2269	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 104   = wceq 1402   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3708  Oncon0 4506  ωcom 4735   × cxp 4770  dom cdm 4772   ↾ cres 4774   Fn wfn 5370   +_o coa 6678  Ncnpi 7633   +_N cpli 7634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-oadd 6685  df-ni 7665  df-pli 7666

This theorem is referenced by: (None)