Theorem dmrecnq 10941

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 10890	. . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q})
2		cnvimass 6075	. . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q
3	1, 2	eqsstri 3985	. . . . 5 ⊢ *Q ⊆ dom ·Q
4		mulnqf 10922	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
5	4	fdmi 6707	. . . . 5 ⊢ dom ·Q = (Q × Q)
6	3, 5	sseqtri 3987	. . . 4 ⊢ *Q ⊆ (Q × Q)
7		dmss 5883	. . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
8	6, 7	ax-mp 5	. . 3 ⊢ dom *Q ⊆ dom (Q × Q)
9		dmxpid 5911	. . 3 ⊢ dom (Q × Q) = Q
10	8, 9	sseqtri 3987	. 2 ⊢ dom *Q ⊆ Q
11		recclnq 10939	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
12		opelxpi 5689	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
13	11, 12	mpdan 699	. . . . . . 7 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
14		df-ov 7403	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩)
15		recidnq 10938	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
16	14, 15	eqtr3id 2814	. . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)
17		ffn 6695	. . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18		fniniseg 7045	. . . . . . . 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 594	. . . . . 6 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}))
21	20, 1	eleqtrrdi 2876	. . . . 5 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
22		df-br 5106	. . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
23	21, 22	sylibr 237	. . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥))
24		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
25		fvex 6884	. . . . 5 ⊢ (*Q‘𝑥) ∈ V
26	24, 25	breldm 5889	. . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q)
27	23, 26	syl 18	. . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q)
28	27	ssriv 3943	. 2 ⊢ Q ⊆ dom *Q
29	10, 28	eqssi 3955	1 ⊢ dom *Q = Q

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ⊆ wss 3907  {csn 4585  ⟨cop 4591   class class class wbr 5105   × cxp 5650  ^◡ccnv 5651  dom cdm 5652   “ cima 5655   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Qcnq 10825  1_Qc1q 10826   ·_Q cmq 10829  *_Qcrq 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-ni 10845  df-mi 10847  df-lti 10848  df-mpq 10882  df-enq 10884  df-nq 10885  df-erq 10886  df-mq 10888  df-1nq 10889  df-rq 10890

This theorem is referenced by:  ltrnq  10952  reclem2pr  11021