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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 9618 | . . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q}) | |
2 | cnvimass 5404 | . . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q | |
3 | 1, 2 | eqsstri 3598 | . . . . 5 ⊢ *Q ⊆ dom ·Q |
4 | mulnqf 9650 | . . . . . 6 ⊢ ·Q :(Q × Q)⟶Q | |
5 | 4 | fdmi 5965 | . . . . 5 ⊢ dom ·Q = (Q × Q) |
6 | 3, 5 | sseqtri 3600 | . . . 4 ⊢ *Q ⊆ (Q × Q) |
7 | dmss 5245 | . . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom *Q ⊆ dom (Q × Q) |
9 | dmxpid 5266 | . . 3 ⊢ dom (Q × Q) = Q | |
10 | 8, 9 | sseqtri 3600 | . 2 ⊢ dom *Q ⊆ Q |
11 | recclnq 9667 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
12 | opelxpi 5072 | . . . . . . . 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 6552 | . . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) | |
15 | recidnq 9666 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
16 | 14, 15 | syl5eqr 2658 | . . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q) |
17 | ffn 5958 | . . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q)) | |
18 | fniniseg 6246 | . . . . . . . 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 695 | . . . . . 6 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q})) |
21 | 20, 1 | syl6eleqr 2699 | . . . . 5 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) |
22 | df-br 4584 | . . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) | |
23 | 21, 22 | sylibr 223 | . . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥)) |
24 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | fvex 6113 | . . . . 5 ⊢ (*Q‘𝑥) ∈ V | |
26 | 24, 25 | breldm 5251 | . . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q) |
27 | 23, 26 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q) |
28 | 27 | ssriv 3572 | . 2 ⊢ Q ⊆ dom *Q |
29 | 10, 28 | eqssi 3584 | 1 ⊢ dom *Q = Q |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 〈cop 4131 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Qcnq 9553 1Qc1q 9554 ·Q cmq 9557 *Qcrq 9558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-mi 9575 df-lti 9576 df-mpq 9610 df-enq 9612 df-nq 9613 df-erq 9614 df-mq 9616 df-1nq 9617 df-rq 9618 |
This theorem is referenced by: ltrnq 9680 reclem2pr 9749 |
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