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Theorem dmrecnq 9669
 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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-rq 9618 . . . . . 6 *Q = ( ·Q “ {1Q})
2 cnvimass 5404 . . . . . 6 ( ·Q “ {1Q}) ⊆ dom ·Q
31, 2eqsstri 3598 . . . . 5 *Q ⊆ dom ·Q
4 mulnqf 9650 . . . . . 6 ·Q :(Q × Q)⟶Q
54fdmi 5965 . . . . 5 dom ·Q = (Q × Q)
63, 5sseqtri 3600 . . . 4 *Q ⊆ (Q × Q)
7 dmss 5245 . . . 4 (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
86, 7ax-mp 5 . . 3 dom *Q ⊆ dom (Q × Q)
9 dmxpid 5266 . . 3 dom (Q × Q) = Q
108, 9sseqtri 3600 . 2 dom *QQ
11 recclnq 9667 . . . . . . . 8 (𝑥Q → (*Q𝑥) ∈ Q)
12 opelxpi 5072 . . . . . . . 8 ((𝑥Q ∧ (*Q𝑥) ∈ Q) → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
1311, 12mpdan 699 . . . . . . 7 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
14 df-ov 6552 . . . . . . . 8 (𝑥 ·Q (*Q𝑥)) = ( ·Q ‘⟨𝑥, (*Q𝑥)⟩)
15 recidnq 9666 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1614, 15syl5eqr 2658 . . . . . . 7 (𝑥Q → ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q)
17 ffn 5958 . . . . . . . 8 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18 fniniseg 6246 . . . . . . . 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 695 . . . . . 6 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}))
2120, 1syl6eleqr 2699 . . . . 5 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
22 df-br 4584 . . . . 5 (𝑥*Q(*Q𝑥) ↔ ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
2321, 22sylibr 223 . . . 4 (𝑥Q𝑥*Q(*Q𝑥))
24 vex 3176 . . . . 5 𝑥 ∈ V
25 fvex 6113 . . . . 5 (*Q𝑥) ∈ V
2624, 25breldm 5251 . . . 4 (𝑥*Q(*Q𝑥) → 𝑥 ∈ dom *Q)
2723, 26syl 17 . . 3 (𝑥Q𝑥 ∈ dom *Q)
2827ssriv 3572 . 2 Q ⊆ dom *Q
2910, 28eqssi 3584 1 dom *Q = Q
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Qcnq 9553  1Qc1q 9554   ·Q cmq 9557  *Qcrq 9558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-mpq 9610  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617  df-rq 9618 This theorem is referenced by:  ltrnq  9680  reclem2pr  9749
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