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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 2737 | . . 3 ⊢ ([Q]‘𝐴) = ([Q]‘𝐴) | |
2 | nqerf 10873 | . . . . 5 ⊢ [Q]:(N × N)⟶Q | |
3 | ffn 6673 | . . . . 5 ⊢ ([Q]:(N × N)⟶Q → [Q] Fn (N × N)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ [Q] Fn (N × N) |
5 | fnbrfvb 6900 | . . . 4 ⊢ (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) | |
6 | 4, 5 | mpan 689 | . . 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 10856 | . . . 4 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
9 | inss1 4193 | . . . 4 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q | |
10 | 8, 9 | eqsstri 3983 | . . 3 ⊢ [Q] ⊆ ~Q |
11 | 10 | ssbri 5155 | . 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 1542 ∈ wcel 2107 ∩ cin 3914 class class class wbr 5110 × cxp 5636 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 Ncnpi 10787 ~Q ceq 10794 Qcnq 10795 [Q]cerq 10797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ni 10815 df-mi 10817 df-lti 10818 df-enq 10854 df-nq 10855 df-erq 10856 df-1nq 10859 |
This theorem is referenced by: nqereq 10878 adderpq 10899 mulerpq 10900 lterpq 10913 |
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