Theorem dmrecnq 10655

Description: Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
dmrecnq	⊢ dom *Q = Q

Proof of Theorem dmrecnq

Step	Hyp	Ref	Expression
1		df-rq 10604	. . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q})
2		cnvimass 5978	. . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q
3	1, 2	eqsstri 3951	. . . . 5 ⊢ *Q ⊆ dom ·Q
4		mulnqf 10636	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
5	4	fdmi 6596	. . . . 5 ⊢ dom ·Q = (Q × Q)
6	3, 5	sseqtri 3953	. . . 4 ⊢ *Q ⊆ (Q × Q)
7		dmss 5800	. . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
8	6, 7	ax-mp 5	. . 3 ⊢ dom *Q ⊆ dom (Q × Q)
9		dmxpid 5828	. . 3 ⊢ dom (Q × Q) = Q
10	8, 9	sseqtri 3953	. 2 ⊢ dom *Q ⊆ Q
11		recclnq 10653	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
12		opelxpi 5617	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
13	11, 12	mpdan 683	. . . . . . 7 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
14		df-ov 7258	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩)
15		recidnq 10652	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
16	14, 15	eqtr3id 2793	. . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)
17		ffn 6584	. . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18		fniniseg 6919	. . . . . . . 8 ⊢ ( ·Q Fn (Q × Q) → (⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}) ↔ (⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)))
19	4, 17, 18	mp2b 10	. . . . . . 7 ⊢ (⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}) ↔ (⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q))
20	13, 16, 19	sylanbrc 582	. . . . . 6 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}))
21	20, 1	eleqtrrdi 2850	. . . . 5 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
22		df-br 5071	. . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
23	21, 22	sylibr 233	. . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥))
24		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
25		fvex 6769	. . . . 5 ⊢ (*Q‘𝑥) ∈ V
26	24, 25	breldm 5806	. . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q)
27	23, 26	syl 17	. . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q)
28	27	ssriv 3921	. 2 ⊢ Q ⊆ dom *Q
29	10, 28	eqssi 3933	1 ⊢ dom *Q = Q

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  {csn 4558  ⟨cop 4564   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579  dom cdm 5580   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Qcnq 10539  1_Qc1q 10540   ·_Q cmq 10543  *_Qcrq 10544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ni 10559  df-mi 10561  df-lti 10562  df-mpq 10596  df-enq 10598  df-nq 10599  df-erq 10600  df-mq 10602  df-1nq 10603  df-rq 10604

This theorem is referenced by:  ltrnq  10666  reclem2pr  10735