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| 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 10970 | . . . . 5 ⊢ [Q]:(N × N)⟶Q | |
| 3 | ffn 6736 | . . . . 5 ⊢ ([Q]:(N × N)⟶Q → [Q] Fn (N × N)) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ [Q] Fn (N × N) | 
| 5 | fnbrfvb 6959 | . . . 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 233 | . 2 ⊢ (𝐴 ∈ (N × N) → 𝐴[Q]([Q]‘𝐴)) | 
| 8 | df-erq 10953 | . . . 4 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q)) | |
| 9 | inss1 4237 | . . . 4 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q | |
| 10 | 8, 9 | eqsstri 4030 | . . 3 ⊢ [Q] ⊆ ~Q | 
| 11 | 10 | ssbri 5188 | . 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 206 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Ncnpi 10884 ~Q ceq 10891 Qcnq 10892 [Q]cerq 10894 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ni 10912 df-mi 10914 df-lti 10915 df-enq 10951 df-nq 10952 df-erq 10953 df-1nq 10956 | 
| This theorem is referenced by: nqereq 10975 adderpq 10996 mulerpq 10997 lterpq 11010 | 
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