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Theorem dmrecnq 9549
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 9498 . . . . . 6 *Q = ( ·Q “ {1Q})
2 cnvimass 5295 . . . . . 6 ( ·Q “ {1Q}) ⊆ dom ·Q
31, 2eqsstri 3502 . . . . 5 *Q ⊆ dom ·Q
4 mulnqf 9530 . . . . . 6 ·Q :(Q × Q)⟶Q
54fdmi 5855 . . . . 5 dom ·Q = (Q × Q)
63, 5sseqtri 3504 . . . 4 *Q ⊆ (Q × Q)
7 dmss 5136 . . . 4 (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
86, 7ax-mp 5 . . 3 dom *Q ⊆ dom (Q × Q)
9 dmxpid 5157 . . 3 dom (Q × Q) = Q
108, 9sseqtri 3504 . 2 dom *QQ
11 recclnq 9547 . . . . . . . 8 (𝑥Q → (*Q𝑥) ∈ Q)
12 opelxpi 4966 . . . . . . . 8 ((𝑥Q ∧ (*Q𝑥) ∈ Q) → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
1311, 12mpdan 698 . . . . . . 7 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ (Q × Q))
14 df-ov 6434 . . . . . . . 8 (𝑥 ·Q (*Q𝑥)) = ( ·Q ‘⟨𝑥, (*Q𝑥)⟩)
15 recidnq 9546 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
1614, 15syl5eqr 2562 . . . . . . 7 (𝑥Q → ( ·Q ‘⟨𝑥, (*Q𝑥)⟩) = 1Q)
17 ffn 5848 . . . . . . . 8 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18 fniniseg 6135 . . . . . . . 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 694 . . . . . 6 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ ( ·Q “ {1Q}))
2120, 1syl6eleqr 2603 . . . . 5 (𝑥Q → ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
22 df-br 4482 . . . . 5 (𝑥*Q(*Q𝑥) ↔ ⟨𝑥, (*Q𝑥)⟩ ∈ *Q)
2321, 22sylibr 222 . . . 4 (𝑥Q𝑥*Q(*Q𝑥))
24 vex 3080 . . . . 5 𝑥 ∈ V
25 fvex 6002 . . . . 5 (*Q𝑥) ∈ V
2624, 25breldm 5142 . . . 4 (𝑥*Q(*Q𝑥) → 𝑥 ∈ dom *Q)
2723, 26syl 17 . . 3 (𝑥Q𝑥 ∈ dom *Q)
2827ssriv 3476 . 2 Q ⊆ dom *Q
2910, 28eqssi 3488 1 dom *Q = Q
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1938  wss 3444  {csn 4028  cop 4034   class class class wbr 4481   × cxp 4930  ccnv 4931  dom cdm 4932  cima 4935   Fn wfn 5689  wf 5690  cfv 5694  (class class class)co 6431  Qcnq 9433  1Qc1q 9434   ·Q cmq 9437  *Qcrq 9438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-ov 6434  df-oprab 6435  df-mpt2 6436  df-om 6839  df-1st 6939  df-2nd 6940  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-1o 7327  df-oadd 7331  df-omul 7332  df-er 7509  df-ni 9453  df-mi 9455  df-lti 9456  df-mpq 9490  df-enq 9492  df-nq 9493  df-erq 9494  df-mq 9496  df-1nq 9497  df-rq 9498
This theorem is referenced by:  ltrnq  9560  reclem2pr  9629
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