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Theorem nqerf 9506
Description: Corollary of nqereu 9505: 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 9489 . . . . . . 7 [Q] = ( ~Q ∩ ((N × N) × Q))
2 inss2 3699 . . . . . . 7 ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
31, 2eqsstri 3502 . . . . . 6 [Q] ⊆ ((N × N) × Q)
4 xpss 5042 . . . . . 6 ((N × N) × Q) ⊆ (V × V)
53, 4sstri 3481 . . . . 5 [Q] ⊆ (V × V)
6 df-rel 4939 . . . . 5 (Rel [Q] ↔ [Q] ⊆ (V × V))
75, 6mpbir 219 . . . 4 Rel [Q]
8 nqereu 9505 . . . . . . . 8 (𝑥 ∈ (N × N) → ∃!𝑦Q 𝑦 ~Q 𝑥)
9 df-reu 2807 . . . . . . . . 9 (∃!𝑦Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦Q𝑦 ~Q 𝑥))
10 eumo 2391 . . . . . . . . 9 (∃!𝑦(𝑦Q𝑦 ~Q 𝑥) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
119, 10sylbi 205 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
128, 11syl 17 . . . . . . 7 (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
13 moanimv 2423 . . . . . . 7 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥)))
1412, 13mpbir 219 . . . . . 6 ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥))
153brel 4984 . . . . . . . . 9 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦Q))
1615simpld 473 . . . . . . . 8 (𝑥[Q]𝑦𝑥 ∈ (N × N))
1715simprd 477 . . . . . . . 8 (𝑥[Q]𝑦𝑦Q)
18 enqer 9497 . . . . . . . . . 10 ~Q Er (N × N)
1918a1i 11 . . . . . . . . 9 (𝑥[Q]𝑦 → ~Q Er (N × N))
20 inss1 3698 . . . . . . . . . . 11 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
211, 20eqsstri 3502 . . . . . . . . . 10 [Q] ⊆ ~Q
2221ssbri 4525 . . . . . . . . 9 (𝑥[Q]𝑦𝑥 ~Q 𝑦)
2319, 22ersym 7516 . . . . . . . 8 (𝑥[Q]𝑦𝑦 ~Q 𝑥)
2416, 17, 23jca32 555 . . . . . . 7 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)))
2524moimi 2412 . . . . . 6 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) → ∃*𝑦 𝑥[Q]𝑦)
2614, 25ax-mp 5 . . . . 5 ∃*𝑦 𝑥[Q]𝑦
2726ax-gen 1700 . . . 4 𝑥∃*𝑦 𝑥[Q]𝑦
28 dffun6 5704 . . . 4 (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
297, 27, 28mpbir2an 956 . . 3 Fun [Q]
30 dmss 5136 . . . . . 6 ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
313, 30ax-mp 5 . . . . 5 dom [Q] ⊆ dom ((N × N) × Q)
32 1nq 9504 . . . . . 6 1QQ
33 ne0i 3783 . . . . . 6 (1QQQ ≠ ∅)
34 dmxp 5156 . . . . . 6 (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
3532, 33, 34mp2b 10 . . . . 5 dom ((N × N) × Q) = (N × N)
3631, 35sseqtri 3504 . . . 4 dom [Q] ⊆ (N × N)
37 reurex 3041 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑦 ~Q 𝑥)
38 simpll 785 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39 simplr 787 . . . . . . . . . . 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 475 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
4240, 41ersym 7516 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
431breqi 4487 . . . . . . . . . . . 12 (𝑥[Q]𝑦𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44 brinxp2 4997 . . . . . . . . . . . 12 (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ (𝑥 ∈ (N × N) ∧ 𝑦Q𝑥 ~Q 𝑦))
4543, 44bitri 262 . . . . . . . . . . 11 (𝑥[Q]𝑦 ↔ (𝑥 ∈ (N × N) ∧ 𝑦Q𝑥 ~Q 𝑦))
4638, 39, 42, 45syl3anbrc 1238 . . . . . . . . . 10 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
4746ex 448 . . . . . . . . 9 ((𝑥 ∈ (N × N) ∧ 𝑦Q) → (𝑦 ~Q 𝑥𝑥[Q]𝑦))
4847reximdva 2904 . . . . . . . 8 (𝑥 ∈ (N × N) → (∃𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑥[Q]𝑦))
49 rexex 2889 . . . . . . . 8 (∃𝑦Q 𝑥[Q]𝑦 → ∃𝑦 𝑥[Q]𝑦)
5037, 48, 49syl56 35 . . . . . . 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 3080 . . . . . . 7 𝑥 ∈ V
5352eldm 5134 . . . . . 6 (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
5451, 53sylibr 222 . . . . 5 (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
5554ssriv 3476 . . . 4 (N × N) ⊆ dom [Q]
5636, 55eqssi 3488 . . 3 dom [Q] = (N × N)
57 df-fn 5692 . . 3 ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
5829, 56, 57mpbir2an 956 . 2 [Q] Fn (N × N)
59 rnss 5166 . . . 4 ([Q] ⊆ ((N × N) × Q) → ran [Q] ⊆ ran ((N × N) × Q))
603, 59ax-mp 5 . . 3 ran [Q] ⊆ ran ((N × N) × Q)
61 rnxpss 5375 . . 3 ran ((N × N) × Q) ⊆ Q
6260, 61sstri 3481 . 2 ran [Q] ⊆ Q
63 df-f 5693 . 2 ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
6458, 62, 63mpbir2an 956 1 [Q]:(N × N)⟶Q
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wal 1472   = wceq 1474  wex 1694  wcel 1938  ∃!weu 2362  ∃*wmo 2363  wne 2684  wrex 2801  ∃!wreu 2802  Vcvv 3077  cin 3443  wss 3444  c0 3777   class class class wbr 4481   × cxp 4930  dom cdm 4932  ran crn 4933  Rel wrel 4937  Fun wfun 5683   Fn wfn 5684  wf 5685   Er wer 7501  Ncnpi 9420   ~Q ceq 9427  Qcnq 9428  1Qc1q 9429  [Q]cerq 9430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-1st 6933  df-2nd 6934  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-1o 7322  df-oadd 7326  df-omul 7327  df-er 7504  df-ni 9448  df-mi 9450  df-lti 9451  df-enq 9487  df-nq 9488  df-erq 9489  df-1nq 9492
This theorem is referenced by:  nqercl  9507  nqerrel  9508  nqerid  9509  addnqf  9524  mulnqf  9525  adderpq  9532  mulerpq  9533  lterpq  9546
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