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Mirrors > Home > MPE Home > Th. List > nqerid | Structured version Visualization version GIF version |
Description: Corollary of nqereu 10923: the function [Q] acts as the identity on members of Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nqerid | ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqerf 10924 | . . 3 ⊢ [Q]:(N × N)⟶Q | |
2 | ffun 6720 | . . 3 ⊢ ([Q]:(N × N)⟶Q → Fun [Q]) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun [Q] |
4 | elpqn 10919 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ Q) | |
6 | enqer 10915 | . . . . 5 ⊢ ~Q Er (N × N) | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ Q → ~Q Er (N × N)) |
8 | 7, 4 | erref 8722 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ~Q 𝐴) |
9 | df-erq 10907 | . . . . 5 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
10 | 9 | breqi 5154 | . . . 4 ⊢ (𝐴[Q]𝐴 ↔ 𝐴( ~Q ∩ ((N × N) × Q))𝐴) |
11 | brinxp2 5753 | . . . 4 ⊢ (𝐴( ~Q ∩ ((N × N) × Q))𝐴 ↔ ((𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q) ∧ 𝐴 ~Q 𝐴)) | |
12 | 10, 11 | bitri 274 | . . 3 ⊢ (𝐴[Q]𝐴 ↔ ((𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q) ∧ 𝐴 ~Q 𝐴)) |
13 | 4, 5, 8, 12 | syl21anbrc 1344 | . 2 ⊢ (𝐴 ∈ Q → 𝐴[Q]𝐴) |
14 | funbrfv 6942 | . 2 ⊢ (Fun [Q] → (𝐴[Q]𝐴 → ([Q]‘𝐴) = 𝐴)) | |
15 | 3, 13, 14 | mpsyl 68 | 1 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3947 class class class wbr 5148 × cxp 5674 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 Er wer 8699 Ncnpi 10838 ~Q ceq 10845 Qcnq 10846 [Q]cerq 10848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-ni 10866 df-mi 10868 df-lti 10869 df-enq 10905 df-nq 10906 df-erq 10907 df-1nq 10910 |
This theorem is referenced by: addassnq 10952 mulassnq 10953 distrnq 10955 mulidnq 10957 recmulnq 10958 1lt2nq 10967 ltexnq 10969 prlem934 11027 |
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