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Mirrors > Home > MPE Home > Th. List > nqerid | Structured version Visualization version GIF version |
Description: Corollary of nqereu 10953: 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 10954 | . . 3 ⊢ [Q]:(N × N)⟶Q | |
2 | ffun 6725 | . . 3 ⊢ ([Q]:(N × N)⟶Q → Fun [Q]) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun [Q] |
4 | elpqn 10949 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ∈ Q) | |
6 | enqer 10945 | . . . . 5 ⊢ ~Q Er (N × N) | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ Q → ~Q Er (N × N)) |
8 | 7, 4 | erref 8745 | . . 3 ⊢ (𝐴 ∈ Q → 𝐴 ~Q 𝐴) |
9 | df-erq 10937 | . . . . 5 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
10 | 9 | breqi 5154 | . . . 4 ⊢ (𝐴[Q]𝐴 ↔ 𝐴( ~Q ∩ ((N × N) × Q))𝐴) |
11 | brinxp2 5755 | . . . 4 ⊢ (𝐴( ~Q ∩ ((N × N) × Q))𝐴 ↔ ((𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q) ∧ 𝐴 ~Q 𝐴)) | |
12 | 10, 11 | bitri 275 | . . 3 ⊢ (𝐴[Q]𝐴 ↔ ((𝐴 ∈ (N × N) ∧ 𝐴 ∈ Q) ∧ 𝐴 ~Q 𝐴)) |
13 | 4, 5, 8, 12 | syl21anbrc 1342 | . 2 ⊢ (𝐴 ∈ Q → 𝐴[Q]𝐴) |
14 | funbrfv 6948 | . 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 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 class class class wbr 5148 × cxp 5676 Fun wfun 6542 ⟶wf 6544 ‘cfv 6548 Er wer 8722 Ncnpi 10868 ~Q ceq 10875 Qcnq 10876 [Q]cerq 10878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ni 10896 df-mi 10898 df-lti 10899 df-enq 10935 df-nq 10936 df-erq 10937 df-1nq 10940 |
This theorem is referenced by: addassnq 10982 mulassnq 10983 distrnq 10985 mulidnq 10987 recmulnq 10988 1lt2nq 10997 ltexnq 10999 prlem934 11057 |
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