Theorem dmaddpi 7287

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 4912	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 6426	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 5297	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 3325	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2191	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 7267	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 4812	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 7266	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3253	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3179	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4597	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3156	. . . . 5 ⊢ N ⊆ On
14		anidm 394	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 145	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4718	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3135	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 144	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4i 2201	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 103   = wceq 1348   ∖ cdif 3118   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414  {csn 3583  Oncon0 4348  ωcom 4574   × cxp 4609  dom cdm 4611   ↾ cres 4613   Fn wfn 5193   +_o coa 6392  Ncnpi 7234   +_N cpli 7235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-ni 7266  df-pli 7267

This theorem is referenced by: (None)