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Mirrors > Home > MPE Home > Th. List > dmrecnq | Structured version Visualization version GIF version |
Description: Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmrecnq | ⊢ dom *Q = Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rq 10076 | . . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q}) | |
2 | cnvimass 5741 | . . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q | |
3 | 1, 2 | eqsstri 3854 | . . . . 5 ⊢ *Q ⊆ dom ·Q |
4 | mulnqf 10108 | . . . . . 6 ⊢ ·Q :(Q × Q)⟶Q | |
5 | 4 | fdmi 6303 | . . . . 5 ⊢ dom ·Q = (Q × Q) |
6 | 3, 5 | sseqtri 3856 | . . . 4 ⊢ *Q ⊆ (Q × Q) |
7 | dmss 5570 | . . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom *Q ⊆ dom (Q × Q) |
9 | dmxpid 5592 | . . 3 ⊢ dom (Q × Q) = Q | |
10 | 8, 9 | sseqtri 3856 | . 2 ⊢ dom *Q ⊆ Q |
11 | recclnq 10125 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
12 | opelxpi 5394 | . . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) | |
13 | 11, 12 | mpdan 677 | . . . . . . 7 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) |
14 | df-ov 6927 | . . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) | |
15 | recidnq 10124 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
16 | 14, 15 | syl5eqr 2828 | . . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q) |
17 | ffn 6293 | . . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q)) | |
18 | fniniseg 6604 | . . . . . . . 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 578 | . . . . . 6 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q})) |
21 | 20, 1 | syl6eleqr 2870 | . . . . 5 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) |
22 | df-br 4889 | . . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) | |
23 | 21, 22 | sylibr 226 | . . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥)) |
24 | vex 3401 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | fvex 6461 | . . . . 5 ⊢ (*Q‘𝑥) ∈ V | |
26 | 24, 25 | breldm 5576 | . . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q) |
27 | 23, 26 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q) |
28 | 27 | ssriv 3825 | . 2 ⊢ Q ⊆ dom *Q |
29 | 10, 28 | eqssi 3837 | 1 ⊢ dom *Q = Q |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 {csn 4398 〈cop 4404 class class class wbr 4888 × cxp 5355 ◡ccnv 5356 dom cdm 5357 “ cima 5360 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 Qcnq 10011 1Qc1q 10012 ·Q cmq 10015 *Qcrq 10016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-omul 7850 df-er 8028 df-ni 10031 df-mi 10033 df-lti 10034 df-mpq 10068 df-enq 10070 df-nq 10071 df-erq 10072 df-mq 10074 df-1nq 10075 df-rq 10076 |
This theorem is referenced by: ltrnq 10138 reclem2pr 10207 |
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