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Theorem dmrecnq 9992
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 9941 . . . . . 6 *Q = ( ·Q “ {1Q})
2 cnvimass 5626 . . . . . 6 ( ·Q “ {1Q}) ⊆ dom ·Q
31, 2eqsstri 3784 . . . . 5 *Q ⊆ dom ·Q
4 mulnqf 9973 . . . . . 6 ·Q :(Q × Q)⟶Q
54fdmi 6192 . . . . 5 dom ·Q = (Q × Q)
63, 5sseqtri 3786 . . . 4 *Q ⊆ (Q × Q)
7 dmss 5461 . . . 4 (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
86, 7ax-mp 5 . . 3 dom *Q ⊆ dom (Q × Q)
9 dmxpid 5483 . . 3 dom (Q × Q) = Q
108, 9sseqtri 3786 . 2 dom *QQ
11 recclnq 9990 . . . . . . . 8 (𝑥Q → (*Q𝑥) ∈ Q)
12 opelxpi 5288 . . . . . . . 8 ((𝑥Q ∧ (*Q𝑥) ∈ Q) → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
1311, 12mpdan 667 . . . . . . 7 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
14 df-ov 6796 . . . . . . . 8 (𝑥 ·Q (*Q𝑥)) = ( ·Q ‘⟨𝑥, (*Q𝑥)⟩)
15 recidnq 9989 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1614, 15syl5eqr 2819 . . . . . . 7 (𝑥Q → ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q)
17 ffn 6185 . . . . . . . 8 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18 fniniseg 6481 . . . . . . . 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 572 . . . . . 6 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}))
2120, 1syl6eleqr 2861 . . . . 5 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
22 df-br 4787 . . . . 5 (𝑥*Q(*Q𝑥) ↔ ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
2321, 22sylibr 224 . . . 4 (𝑥Q𝑥*Q(*Q𝑥))
24 vex 3354 . . . . 5 𝑥 ∈ V
25 fvex 6342 . . . . 5 (*Q𝑥) ∈ V
2624, 25breldm 5467 . . . 4 (𝑥*Q(*Q𝑥) → 𝑥 ∈ dom *Q)
2723, 26syl 17 . . 3 (𝑥Q𝑥 ∈ dom *Q)
2827ssriv 3756 . 2 Q ⊆ dom *Q
2910, 28eqssi 3768 1 dom *Q = Q
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wss 3723  {csn 4316  cop 4322   class class class wbr 4786   × cxp 5247  ccnv 5248  dom cdm 5249  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Qcnq 9876  1Qc1q 9877   ·Q cmq 9880  *Qcrq 9881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ni 9896  df-mi 9898  df-lti 9899  df-mpq 9933  df-enq 9935  df-nq 9936  df-erq 9937  df-mq 9939  df-1nq 9940  df-rq 9941
This theorem is referenced by:  ltrnq  10003  reclem2pr  10072
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