Theorem dmrecnq 9478

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.

Step	Hyp	Ref	Expression
1		df-rq 9427	. . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q})
2		cnvimass 5238	. . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q
3	1, 2	eqsstri 3484	. . . . 5 ⊢ *Q ⊆ dom ·Q
4		mulnqf 9459	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
5	4	fdmi 5791	. . . . 5 ⊢ dom ·Q = (Q × Q)
6	3, 5	sseqtri 3486	. . . 4 ⊢ *Q ⊆ (Q × Q)
7		dmss 5082	. . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q))
8	6, 7	ax-mp 5	. . 3 ⊢ dom *Q ⊆ dom (Q × Q)
9		dmxpid 5103	. . 3 ⊢ dom (Q × Q) = Q
10	8, 9	sseqtri 3486	. 2 ⊢ dom *Q ⊆ Q
11		recclnq 9476	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
12		opelxpi 4912	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
13	11, 12	mpdan 690	. . . . . . 7 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (Q × Q))
14		df-ov 6366	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩)
15		recidnq 9475	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
16	14, 15	syl5eqr 2553	. . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘⟨𝑥, (*Q‘𝑥)⟩) = 1Q)
17		ffn 5784	. . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
18		fniniseg 6070	. . . . . . . 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 686	. . . . . 6 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ (◡ ·Q “ {1Q}))
21	20, 1	syl6eleqr 2594	. . . . 5 ⊢ (𝑥 ∈ Q → ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
22		df-br 4435	. . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ ⟨𝑥, (*Q‘𝑥)⟩ ∈ *Q)
23	21, 22	sylibr 219	. . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥))
24		vex 3069	. . . . 5 ⊢ 𝑥 ∈ V
25		fvex 5937	. . . . 5 ⊢ (*Q‘𝑥) ∈ V
26	24, 25	breldm 5088	. . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q)
27	23, 26	syl 17	. . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q)
28	27	ssriv 3458	. 2 ⊢ Q ⊆ dom *Q
29	10, 28	eqssi 3470	1 ⊢ dom *Q = Q

Syntax hints:   ↔ wb 191   ∧ wa 378   = wceq 1468   ∈ wcel 1937   ⊆ wss 3426  {csn 3995  ⟨cop 4001   class class class wbr 4434   × cxp 4878  ^◡ccnv 4879  dom cdm 4880   “ cima 4883   Fn wfn 5628  ⟶wf 5629  ‘cfv 5633  (class class class)co 6363  Qcnq 9362  1_Qc1q 9363   ·_Q cmq 9366  *_Qcrq 9367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659

This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ni 9382  df-mi 9384  df-lti 9385  df-mpq 9419  df-enq 9421  df-nq 9422  df-erq 9423  df-mq 9425  df-1nq 9426  df-rq 9427

This theorem is referenced by:  ltrnq  9489  reclem2pr  9558