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Theorem dmrecnq 10368
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
StepHypRef Expression
1 df-rq 10317 . . . . . 6 *Q = ( ·Q “ {1Q})
2 cnvimass 5925 . . . . . 6 ( ·Q “ {1Q}) ⊆ dom ·Q
31, 2eqsstri 3980 . . . . 5 *Q ⊆ dom ·Q
4 mulnqf 10349 . . . . . 6 ·Q :(Q × Q)⟶Q
54fdmi 6500 . . . . 5 dom ·Q = (Q × Q)
63, 5sseqtri 3982 . . . 4 *Q ⊆ (Q × Q)
7 dmss 5747 . . . 4 (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
86, 7ax-mp 5 . . 3 dom *Q ⊆ dom (Q × Q)
9 dmxpid 5776 . . 3 dom (Q × Q) = Q
108, 9sseqtri 3982 . 2 dom *QQ
11 recclnq 10366 . . . . . . . 8 (𝑥Q → (*Q𝑥) ∈ Q)
12 opelxpi 5568 . . . . . . . 8 ((𝑥Q ∧ (*Q𝑥) ∈ Q) → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
1311, 12mpdan 685 . . . . . . 7 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
14 df-ov 7136 . . . . . . . 8 (𝑥 ·Q (*Q𝑥)) = ( ·Q ‘⟨𝑥, (*Q𝑥)⟩)
15 recidnq 10365 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1614, 15syl5eqr 2869 . . . . . . 7 (𝑥Q → ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q)
17 ffn 6490 . . . . . . . 8 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18 fniniseg 6806 . . . . . . . 8 ( ·Q Fn (Q × Q) → (⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q)))
194, 17, 18mp2b 10 . . . . . . 7 (⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q))
2013, 16, 19sylanbrc 585 . . . . . 6 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}))
2120, 1eleqtrrdi 2922 . . . . 5 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
22 df-br 5043 . . . . 5 (𝑥*Q(*Q𝑥) ↔ ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
2321, 22sylibr 236 . . . 4 (𝑥Q𝑥*Q(*Q𝑥))
24 vex 3476 . . . . 5 𝑥 ∈ V
25 fvex 6659 . . . . 5 (*Q𝑥) ∈ V
2624, 25breldm 5753 . . . 4 (𝑥*Q(*Q𝑥) → 𝑥 ∈ dom *Q)
2723, 26syl 17 . . 3 (𝑥Q𝑥 ∈ dom *Q)
2827ssriv 3950 . 2 Q ⊆ dom *Q
2910, 28eqssi 3962 1 dom *Q = Q
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wcel 2114  wss 3913  {csn 4543  cop 4549   class class class wbr 5042   × cxp 5529  ccnv 5530  dom cdm 5531  cima 5534   Fn wfn 6326  wf 6327  cfv 6331  (class class class)co 7133  Qcnq 10252  1Qc1q 10253   ·Q cmq 10256  *Qcrq 10257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-omul 8085  df-er 8267  df-ni 10272  df-mi 10274  df-lti 10275  df-mpq 10309  df-enq 10311  df-nq 10312  df-erq 10313  df-mq 10315  df-1nq 10316  df-rq 10317
This theorem is referenced by:  ltrnq  10379  reclem2pr  10448
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