Theorem dmaddpi 6423

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 4632	. . 3 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ dom +𝑜 )
2		fnoa 6027	. . . . 5 ⊢ +𝑜 Fn (On × On)
3		fndm 4998	. . . . 5 ⊢ ( +𝑜 Fn (On × On) → dom +𝑜 = (On × On))
4	2, 3	ax-mp 7	. . . 4 ⊢ dom +𝑜 = (On × On)
5	4	ineq2i 3135	. . 3 ⊢ ((N × N) ∩ dom +𝑜 ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2060	. 2 ⊢ dom ( +𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 6403	. . 3 ⊢ +N = ( +𝑜 ↾ (N × N))
8	7	dmeqi 4536	. 2 ⊢ dom +N = dom ( +𝑜 ↾ (N × N))
9		df-ni 6402	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3070	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 2975	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4335	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 2954	. . . . 5 ⊢ N ⊆ On
14		anidm 376	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 134	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4445	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 7	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 2932	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 133	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2070	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 97   = wceq 1243   ∖ cdif 2914   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  {csn 3375  Oncon0 4100  ωcom 4313   × cxp 4343  dom cdm 4345   ↾ cres 4347   Fn wfn 4897   +_𝑜 coa 5998  Ncnpi 6370   +_N cpli 6371

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-ni 6402  df-pli 6403

This theorem is referenced by: (None)