Theorem dmrecnq 10891

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 10840	. . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q})
2		cnvimass 6048	. . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q
3	1, 2	eqsstri 3969	. . . . 5 ⊢ *Q ⊆ dom ·Q
4		mulnqf 10872	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
5	4	fdmi 6680	. . . . 5 ⊢ dom ·Q = (Q × Q)
6	3, 5	sseqtri 3971	. . . 4 ⊢ *Q ⊆ (Q × Q)
7		dmss 5858	. . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
8	6, 7	ax-mp 5	. . 3 ⊢ dom *Q ⊆ dom (Q × Q)
9		dmxpid 5886	. . 3 ⊢ dom (Q × Q) = Q
10	8, 9	sseqtri 3971	. 2 ⊢ dom *Q ⊆ Q
11		recclnq 10889	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
12		opelxpi 5668	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
13	11, 12	mpdan 688	. . . . . . 7 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
14		df-ov 7370	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩)
15		recidnq 10888	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
16	14, 15	eqtr3id 2786	. . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)
17		ffn 6669	. . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18		fniniseg 7013	. . . . . . . 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 584	. . . . . 6 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}))
21	20, 1	eleqtrrdi 2848	. . . . 5 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
22		df-br 5087	. . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
23	21, 22	sylibr 234	. . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥))
24		vex 3434	. . . . 5 ⊢ 𝑥 ∈ V
25		fvex 6854	. . . . 5 ⊢ (*Q‘𝑥) ∈ V
26	24, 25	breldm 5864	. . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q)
27	23, 26	syl 17	. . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q)
28	27	ssriv 3926	. 2 ⊢ Q ⊆ dom *Q
29	10, 28	eqssi 3939	1 ⊢ dom *Q = Q

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  {csn 4568  ⟨cop 4574   class class class wbr 5086   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7367  Qcnq 10775  1_Qc1q 10776   ·_Q cmq 10779  *_Qcrq 10780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-ni 10795  df-mi 10797  df-lti 10798  df-mpq 10832  df-enq 10834  df-nq 10835  df-erq 10836  df-mq 10838  df-1nq 10839  df-rq 10840

This theorem is referenced by:  ltrnq  10902  reclem2pr  10971