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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 2736 | . . 3 ⊢ ([Q]‘𝐴) = ([Q]‘𝐴) | |
2 | nqerf 10509 | . . . . 5 ⊢ [Q]:(N × N)⟶Q | |
3 | ffn 6523 | . . . . 5 ⊢ ([Q]:(N × N)⟶Q → [Q] Fn (N × N)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ [Q] Fn (N × N) |
5 | fnbrfvb 6743 | . . . 4 ⊢ (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) | |
6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐴 ∈ (N × N) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴))) |
7 | 1, 6 | mpbii 236 | . 2 ⊢ (𝐴 ∈ (N × N) → 𝐴[Q]([Q]‘𝐴)) |
8 | df-erq 10492 | . . . 4 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
9 | inss1 4129 | . . . 4 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q | |
10 | 8, 9 | eqsstri 3921 | . . 3 ⊢ [Q] ⊆ ~Q |
11 | 10 | ssbri 5084 | . 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 209 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 class class class wbr 5039 × cxp 5534 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 Ncnpi 10423 ~Q ceq 10430 Qcnq 10431 [Q]cerq 10433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ni 10451 df-mi 10453 df-lti 10454 df-enq 10490 df-nq 10491 df-erq 10492 df-1nq 10495 |
This theorem is referenced by: nqereq 10514 adderpq 10535 mulerpq 10536 lterpq 10549 |
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