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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 2771 | . . 3 ⊢ ([Q]‘𝐴) = ([Q]‘𝐴) | |
2 | nqerf 9958 | . . . . 5 ⊢ [Q]:(N × N)⟶Q | |
3 | ffn 6184 | . . . . 5 ⊢ ([Q]:(N × N)⟶Q → [Q] Fn (N × N)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ [Q] Fn (N × N) |
5 | fnbrfvb 6379 | . . . 4 ⊢ (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) | |
6 | 4, 5 | mpan 670 | . . 3 ⊢ (𝐴 ∈ (N × N) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) |
7 | 1, 6 | mpbii 223 | . 2 ⊢ (𝐴 ∈ (N × N) → 𝐴[Q]([Q]‘𝐴)) |
8 | df-erq 9941 | . . . 4 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
9 | inss1 3981 | . . . 4 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q | |
10 | 8, 9 | eqsstri 3784 | . . 3 ⊢ [Q] ⊆ ~Q |
11 | 10 | ssbri 4832 | . 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 196 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 class class class wbr 4787 × cxp 5248 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 Ncnpi 9872 ~Q ceq 9879 Qcnq 9880 [Q]cerq 9882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-omul 7722 df-er 7900 df-ni 9900 df-mi 9902 df-lti 9903 df-enq 9939 df-nq 9940 df-erq 9941 df-1nq 9944 |
This theorem is referenced by: nqereq 9963 adderpq 9984 mulerpq 9985 lterpq 9998 |
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