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Mirrors > Home > MPE Home > Th. List > nqerid | Structured version Visualization version GIF version |
Description: Corollary of nqereu 9630: 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 9631 | . . 3 ⊢ [Q]:(N × N)⟶Q | |
2 | ffun 5961 | . . 3 ⊢ ([Q]:(N × N)⟶Q → Fun [Q]) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun [Q] |
4 | elpqn 9626 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ Q) | |
6 | enqer 9622 | . . . . 5 ⊢ ~Q Er (N × N) | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ Q → ~Q Er (N × N)) |
8 | 7, 4 | erref 7649 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ~Q 𝐴) |
9 | df-erq 9614 | . . . . 5 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
10 | 9 | breqi 4589 | . . . 4 ⊢ (𝐴[Q]𝐴 ↔ 𝐴( ~Q ∩ ((N × N) × Q))𝐴) |
11 | brinxp2 5103 | . . . 4 ⊢ (𝐴( ~Q ∩ ((N × N) × Q))𝐴 ↔ (𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q ∧ 𝐴 ~Q 𝐴)) | |
12 | 10, 11 | bitri 263 | . . 3 ⊢ (𝐴[Q]𝐴 ↔ (𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q ∧ 𝐴 ~Q 𝐴)) |
13 | 4, 5, 8, 12 | syl3anbrc 1239 | . 2 ⊢ (𝐴 ∈ Q → 𝐴[Q]𝐴) |
14 | funbrfv 6144 | . 2 ⊢ (Fun [Q] → (𝐴[Q]𝐴 → ([Q]‘𝐴) = 𝐴)) | |
15 | 3, 13, 14 | mpsyl 66 | 1 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 class class class wbr 4583 × cxp 5036 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 Er wer 7626 Ncnpi 9545 ~Q ceq 9552 Qcnq 9553 [Q]cerq 9555 |
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-enq 9612 df-nq 9613 df-erq 9614 df-1nq 9617 |
This theorem is referenced by: addassnq 9659 mulassnq 9660 distrnq 9662 mulidnq 9664 recmulnq 9665 1lt2nq 9674 ltexnq 9676 prlem934 9734 |
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