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Theorem nqerf 9955
Description: Corollary of nqereu 9954: the function [Q] is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqerf [Q]:(N × N)⟶Q

Proof of Theorem nqerf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-erq 9938 . . . . . . 7 [Q] = ( ~Q ∩ ((N × N) × Q))
2 inss2 3983 . . . . . . 7 ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
31, 2eqsstri 3785 . . . . . 6 [Q] ⊆ ((N × N) × Q)
4 xpss 5266 . . . . . 6 ((N × N) × Q) ⊆ (V × V)
53, 4sstri 3762 . . . . 5 [Q] ⊆ (V × V)
6 df-rel 5257 . . . . 5 (Rel [Q] ↔ [Q] ⊆ (V × V))
75, 6mpbir 221 . . . 4 Rel [Q]
8 nqereu 9954 . . . . . . . 8 (𝑥 ∈ (N × N) → ∃!𝑦Q 𝑦 ~Q 𝑥)
9 df-reu 3068 . . . . . . . . 9 (∃!𝑦Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦Q𝑦 ~Q 𝑥))
10 eumo 2647 . . . . . . . . 9 (∃!𝑦(𝑦Q𝑦 ~Q 𝑥) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
119, 10sylbi 207 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
128, 11syl 17 . . . . . . 7 (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
13 moanimv 2680 . . . . . . 7 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥)))
1412, 13mpbir 221 . . . . . 6 ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥))
153brel 5309 . . . . . . . . 9 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦Q))
1615simpld 478 . . . . . . . 8 (𝑥[Q]𝑦𝑥 ∈ (N × N))
1715simprd 479 . . . . . . . 8 (𝑥[Q]𝑦𝑦Q)
18 enqer 9946 . . . . . . . . . 10 ~Q Er (N × N)
1918a1i 11 . . . . . . . . 9 (𝑥[Q]𝑦 → ~Q Er (N × N))
20 inss1 3982 . . . . . . . . . . 11 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
211, 20eqsstri 3785 . . . . . . . . . 10 [Q] ⊆ ~Q
2221ssbri 4832 . . . . . . . . 9 (𝑥[Q]𝑦𝑥 ~Q 𝑦)
2319, 22ersym 7909 . . . . . . . 8 (𝑥[Q]𝑦𝑦 ~Q 𝑥)
2416, 17, 23jca32 501 . . . . . . 7 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)))
2524moimi 2669 . . . . . 6 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) → ∃*𝑦 𝑥[Q]𝑦)
2614, 25ax-mp 5 . . . . 5 ∃*𝑦 𝑥[Q]𝑦
2726ax-gen 1870 . . . 4 𝑥∃*𝑦 𝑥[Q]𝑦
28 dffun6 6047 . . . 4 (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
297, 27, 28mpbir2an 684 . . 3 Fun [Q]
30 dmss 5462 . . . . . 6 ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
313, 30ax-mp 5 . . . . 5 dom [Q] ⊆ dom ((N × N) × Q)
32 1nq 9953 . . . . . 6 1QQ
33 ne0i 4070 . . . . . 6 (1QQQ ≠ ∅)
34 dmxp 5483 . . . . . 6 (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
3532, 33, 34mp2b 10 . . . . 5 dom ((N × N) × Q) = (N × N)
3631, 35sseqtri 3787 . . . 4 dom [Q] ⊆ (N × N)
37 reurex 3309 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑦 ~Q 𝑥)
38 simpll 744 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39 simplr 746 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑦Q)
4018a1i 11 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → ~Q Er (N × N))
41 simpr 471 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
4240, 41ersym 7909 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
431breqi 4793 . . . . . . . . . . . 12 (𝑥[Q]𝑦𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44 brinxp2 5321 . . . . . . . . . . . 12 (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ (𝑥 ∈ (N × N) ∧ 𝑦Q𝑥 ~Q 𝑦))
4543, 44bitri 264 . . . . . . . . . . 11 (𝑥[Q]𝑦 ↔ (𝑥 ∈ (N × N) ∧ 𝑦Q𝑥 ~Q 𝑦))
4638, 39, 42, 45syl3anbrc 1428 . . . . . . . . . 10 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
4746ex 397 . . . . . . . . 9 ((𝑥 ∈ (N × N) ∧ 𝑦Q) → (𝑦 ~Q 𝑥𝑥[Q]𝑦))
4847reximdva 3165 . . . . . . . 8 (𝑥 ∈ (N × N) → (∃𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑥[Q]𝑦))
49 rexex 3150 . . . . . . . 8 (∃𝑦Q 𝑥[Q]𝑦 → ∃𝑦 𝑥[Q]𝑦)
5037, 48, 49syl56 36 . . . . . . 7 (𝑥 ∈ (N × N) → (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦 𝑥[Q]𝑦))
518, 50mpd 15 . . . . . 6 (𝑥 ∈ (N × N) → ∃𝑦 𝑥[Q]𝑦)
52 vex 3354 . . . . . . 7 𝑥 ∈ V
5352eldm 5460 . . . . . 6 (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
5451, 53sylibr 224 . . . . 5 (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
5554ssriv 3757 . . . 4 (N × N) ⊆ dom [Q]
5636, 55eqssi 3769 . . 3 dom [Q] = (N × N)
57 df-fn 6035 . . 3 ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
5829, 56, 57mpbir2an 684 . 2 [Q] Fn (N × N)
59 rnss 5493 . . . 4 ([Q] ⊆ ((N × N) × Q) → ran [Q] ⊆ ran ((N × N) × Q))
603, 59ax-mp 5 . . 3 ran [Q] ⊆ ran ((N × N) × Q)
61 rnxpss 5708 . . 3 ran ((N × N) × Q) ⊆ Q
6260, 61sstri 3762 . 2 ran [Q] ⊆ Q
63 df-f 6036 . 2 ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
6458, 62, 63mpbir2an 684 1 [Q]:(N × N)⟶Q
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  ∃!weu 2618  ∃*wmo 2619  wne 2943  wrex 3062  ∃!wreu 3063  Vcvv 3351  cin 3723  wss 3724  c0 4064   class class class wbr 4787   × cxp 5248  dom cdm 5250  ran crn 5251  Rel wrel 5255  Fun wfun 6026   Fn wfn 6027  wf 6028   Er wer 7894  Ncnpi 9869   ~Q ceq 9876  Qcnq 9877  1Qc1q 9878  [Q]cerq 9879
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 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  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 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-omul 7719  df-er 7897  df-ni 9897  df-mi 9899  df-lti 9900  df-enq 9936  df-nq 9937  df-erq 9938  df-1nq 9941
This theorem is referenced by:  nqercl  9956  nqerrel  9957  nqerid  9958  addnqf  9973  mulnqf  9974  adderpq  9981  mulerpq  9982  lterpq  9995
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