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Mirrors > Home > MPE Home > Th. List > nqerrel | Structured version Visualization version GIF version |
Description: Any member of (N × N) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nqerrel | ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ ([Q]‘𝐴) = ([Q]‘𝐴) | |
2 | nqerf 10973 | . . . . 5 ⊢ [Q]:(N × N)⟶Q | |
3 | ffn 6728 | . . . . 5 ⊢ ([Q]:(N × N)⟶Q → [Q] Fn (N × N)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ [Q] Fn (N × N) |
5 | fnbrfvb 6954 | . . . 4 ⊢ (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) | |
6 | 4, 5 | mpan 688 | . . 3 ⊢ (𝐴 ∈ (N × N) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) |
7 | 1, 6 | mpbii 232 | . 2 ⊢ (𝐴 ∈ (N × N) → 𝐴[Q]([Q]‘𝐴)) |
8 | df-erq 10956 | . . . 4 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
9 | inss1 4230 | . . . 4 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q | |
10 | 8, 9 | eqsstri 4014 | . . 3 ⊢ [Q] ⊆ ~Q |
11 | 10 | ssbri 5198 | . 2 ⊢ (𝐴[Q]([Q]‘𝐴) → 𝐴 ~Q ([Q]‘𝐴)) |
12 | 7, 11 | syl 17 | 1 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 class class class wbr 5153 × cxp 5680 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 Ncnpi 10887 ~Q ceq 10894 Qcnq 10895 [Q]cerq 10897 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-omul 8501 df-er 8734 df-ni 10915 df-mi 10917 df-lti 10918 df-enq 10954 df-nq 10955 df-erq 10956 df-1nq 10959 |
This theorem is referenced by: nqereq 10978 adderpq 10999 mulerpq 11000 lterpq 11013 |
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