Theorem dmmulpi 7459

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

Assertion

Ref	Expression
dmmulpi	⊢ dom ·N = (N × N)

Proof of Theorem dmmulpi

Step	Hyp	Ref	Expression
1		dmres 4989	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 6549	. . . . 5 ⊢ ·o Fn (On × On)
3		fndm 5382	. . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
4	2, 3	ax-mp 5	. . . 4 ⊢ dom ·o = (On × On)
5	4	ineq2i 3375	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtri 2227	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-mi 7439	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
8	7	dmeqi 4888	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
9		df-ni 7437	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 3303	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstri 3229	. . . . . 6 ⊢ N ⊆ ω
12		omsson 4669	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 3206	. . . . 5 ⊢ N ⊆ On
14		anidm 396	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 146	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		xpss12 4790	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 3184	. . 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 2237	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 104   = wceq 1373   ∖ cdif 3167   ∩ cin 3169   ⊆ wss 3170  ∅c0 3464  {csn 3638  Oncon0 4418  ωcom 4646   × cxp 4681  dom cdm 4683   ↾ cres 4685   Fn wfn 5275   ·_o comu 6513  Ncnpi 7405   ·_N cmi 7407

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-oadd 6519  df-omul 6520  df-ni 7437  df-mi 7439

This theorem is referenced by: (None)