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Theorem nqerf 10350
Description: Corollary of nqereu 10349: 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 10333 . . . . . . 7 [Q] = ( ~Q ∩ ((N × N) × Q))
2 inss2 4191 . . . . . . 7 ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
31, 2eqsstri 3987 . . . . . 6 [Q] ⊆ ((N × N) × Q)
4 xpss 5558 . . . . . 6 ((N × N) × Q) ⊆ (V × V)
53, 4sstri 3962 . . . . 5 [Q] ⊆ (V × V)
6 df-rel 5549 . . . . 5 (Rel [Q] ↔ [Q] ⊆ (V × V))
75, 6mpbir 234 . . . 4 Rel [Q]
8 nqereu 10349 . . . . . . . 8 (𝑥 ∈ (N × N) → ∃!𝑦Q 𝑦 ~Q 𝑥)
9 df-reu 3140 . . . . . . . . 9 (∃!𝑦Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦Q𝑦 ~Q 𝑥))
10 eumo 2664 . . . . . . . . 9 (∃!𝑦(𝑦Q𝑦 ~Q 𝑥) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
119, 10sylbi 220 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
128, 11syl 17 . . . . . . 7 (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
13 moanimv 2707 . . . . . . 7 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥)))
1412, 13mpbir 234 . . . . . 6 ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥))
153brel 5604 . . . . . . . . 9 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦Q))
1615simpld 498 . . . . . . . 8 (𝑥[Q]𝑦𝑥 ∈ (N × N))
1715simprd 499 . . . . . . . 8 (𝑥[Q]𝑦𝑦Q)
18 enqer 10341 . . . . . . . . . 10 ~Q Er (N × N)
1918a1i 11 . . . . . . . . 9 (𝑥[Q]𝑦 → ~Q Er (N × N))
20 inss1 4190 . . . . . . . . . . 11 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
211, 20eqsstri 3987 . . . . . . . . . 10 [Q] ⊆ ~Q
2221ssbri 5097 . . . . . . . . 9 (𝑥[Q]𝑦𝑥 ~Q 𝑦)
2319, 22ersym 8297 . . . . . . . 8 (𝑥[Q]𝑦𝑦 ~Q 𝑥)
2416, 17, 23jca32 519 . . . . . . 7 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)))
2524moimi 2629 . . . . . 6 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) → ∃*𝑦 𝑥[Q]𝑦)
2614, 25ax-mp 5 . . . . 5 ∃*𝑦 𝑥[Q]𝑦
2726ax-gen 1797 . . . 4 𝑥∃*𝑦 𝑥[Q]𝑦
28 dffun6 6358 . . . 4 (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
297, 27, 28mpbir2an 710 . . 3 Fun [Q]
30 dmss 5758 . . . . . 6 ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
313, 30ax-mp 5 . . . . 5 dom [Q] ⊆ dom ((N × N) × Q)
32 1nq 10348 . . . . . 6 1QQ
33 ne0i 4283 . . . . . 6 (1QQQ ≠ ∅)
34 dmxp 5786 . . . . . 6 (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
3532, 33, 34mp2b 10 . . . . 5 dom ((N × N) × Q) = (N × N)
3631, 35sseqtri 3989 . . . 4 dom [Q] ⊆ (N × N)
37 reurex 3414 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑦 ~Q 𝑥)
38 simpll 766 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39 simplr 768 . . . . . . . . . . 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 488 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
4240, 41ersym 8297 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
431breqi 5058 . . . . . . . . . . . 12 (𝑥[Q]𝑦𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44 brinxp2 5616 . . . . . . . . . . . 12 (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑥 ~Q 𝑦))
4543, 44bitri 278 . . . . . . . . . . 11 (𝑥[Q]𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑥 ~Q 𝑦))
4638, 39, 42, 45syl21anbrc 1341 . . . . . . . . . 10 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
4746ex 416 . . . . . . . . 9 ((𝑥 ∈ (N × N) ∧ 𝑦Q) → (𝑦 ~Q 𝑥𝑥[Q]𝑦))
4847reximdva 3266 . . . . . . . 8 (𝑥 ∈ (N × N) → (∃𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑥[Q]𝑦))
49 rexex 3234 . . . . . . . 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 3483 . . . . . . 7 𝑥 ∈ V
5352eldm 5756 . . . . . 6 (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
5451, 53sylibr 237 . . . . 5 (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
5554ssriv 3957 . . . 4 (N × N) ⊆ dom [Q]
5636, 55eqssi 3969 . . 3 dom [Q] = (N × N)
57 df-fn 6346 . . 3 ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
5829, 56, 57mpbir2an 710 . 2 [Q] Fn (N × N)
593rnssi 5797 . . 3 ran [Q] ⊆ ran ((N × N) × Q)
60 rnxpss 6016 . . 3 ran ((N × N) × Q) ⊆ Q
6159, 60sstri 3962 . 2 ran [Q] ⊆ Q
62 df-f 6347 . 2 ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
6358, 61, 62mpbir2an 710 1 [Q]:(N × N)⟶Q
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  ∃*wmo 2622  ∃!weu 2654  wne 3014  wrex 3134  ∃!wreu 3135  Vcvv 3480  cin 3918  wss 3919  c0 4276   class class class wbr 5052   × cxp 5540  dom cdm 5542  ran crn 5543  Rel wrel 5547  Fun wfun 6337   Fn wfn 6338  wf 6339   Er wer 8282  Ncnpi 10264   ~Q ceq 10271  Qcnq 10272  1Qc1q 10273  [Q]cerq 10274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-er 8285  df-ni 10292  df-mi 10294  df-lti 10295  df-enq 10331  df-nq 10332  df-erq 10333  df-1nq 10336
This theorem is referenced by:  nqercl  10351  nqerrel  10352  nqerid  10353  addnqf  10368  mulnqf  10369  adderpq  10376  mulerpq  10377  lterpq  10390
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