Theorem dmrecnq 9790

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 9739	. . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q})
2		cnvimass 5485	. . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q
3	1, 2	eqsstri 3635	. . . . 5 ⊢ *Q ⊆ dom ·Q
4		mulnqf 9771	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
5	4	fdmi 6052	. . . . 5 ⊢ dom ·Q = (Q × Q)
6	3, 5	sseqtri 3637	. . . 4 ⊢ *Q ⊆ (Q × Q)
7		dmss 5323	. . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
8	6, 7	ax-mp 5	. . 3 ⊢ dom *Q ⊆ dom (Q × Q)
9		dmxpid 5345	. . 3 ⊢ dom (Q × Q) = Q
10	8, 9	sseqtri 3637	. 2 ⊢ dom *Q ⊆ Q
11		recclnq 9788	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
12		opelxpi 5148	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
13	11, 12	mpdan 702	. . . . . . 7 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
14		df-ov 6653	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩)
15		recidnq 9787	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
16	14, 15	syl5eqr 2670	. . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)
17		ffn 6045	. . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18		fniniseg 6338	. . . . . . . 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 698	. . . . . 6 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}))
21	20, 1	syl6eleqr 2712	. . . . 5 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
22		df-br 4654	. . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
23	21, 22	sylibr 224	. . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥))
24		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
25		fvex 6201	. . . . 5 ⊢ (*Q‘𝑥) ∈ V
26	24, 25	breldm 5329	. . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q)
27	23, 26	syl 17	. . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q)
28	27	ssriv 3607	. 2 ⊢ Q ⊆ dom *Q
29	10, 28	eqssi 3619	1 ⊢ dom *Q = Q

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Qcnq 9674  1_Qc1q 9675   ·_Q cmq 9678  *_Qcrq 9679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-mpq 9731  df-enq 9733  df-nq 9734  df-erq 9735  df-mq 9737  df-1nq 9738  df-rq 9739

This theorem is referenced by:  ltrnq  9801  reclem2pr  9870