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Theorem nqerf 9607
Description: Corollary of nqereu 9606: 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 9590 . . . . . . 7 [Q] = ( ~Q ∩ ((N × N) × Q))
2 inss2 3794 . . . . . . 7 ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
31, 2eqsstri 3596 . . . . . 6 [Q] ⊆ ((N × N) × Q)
4 xpss 5137 . . . . . 6 ((N × N) × Q) ⊆ (V × V)
53, 4sstri 3575 . . . . 5 [Q] ⊆ (V × V)
6 df-rel 5034 . . . . 5 (Rel [Q] ↔ [Q] ⊆ (V × V))
75, 6mpbir 219 . . . 4 Rel [Q]
8 nqereu 9606 . . . . . . . 8 (𝑥 ∈ (N × N) → ∃!𝑦Q 𝑦 ~Q 𝑥)
9 df-reu 2901 . . . . . . . . 9 (∃!𝑦Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦Q𝑦 ~Q 𝑥))
10 eumo 2485 . . . . . . . . 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 2517 . . . . . . 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 5079 . . . . . . . . 9 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦Q))
1615simpld 473 . . . . . . . 8 (𝑥[Q]𝑦𝑥 ∈ (N × N))
1715simprd 477 . . . . . . . 8 (𝑥[Q]𝑦𝑦Q)
18 enqer 9598 . . . . . . . . . 10 ~Q Er (N × N)
1918a1i 11 . . . . . . . . 9 (𝑥[Q]𝑦 → ~Q Er (N × N))
20 inss1 3793 . . . . . . . . . . 11 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
211, 20eqsstri 3596 . . . . . . . . . 10 [Q] ⊆ ~Q
2221ssbri 4620 . . . . . . . . 9 (𝑥[Q]𝑦𝑥 ~Q 𝑦)
2319, 22ersym 7617 . . . . . . . 8 (𝑥[Q]𝑦𝑦 ~Q 𝑥)
2416, 17, 23jca32 555 . . . . . . 7 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)))
2524moimi 2506 . . . . . 6 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) → ∃*𝑦 𝑥[Q]𝑦)
2614, 25ax-mp 5 . . . . 5 ∃*𝑦 𝑥[Q]𝑦
2726ax-gen 1712 . . . 4 𝑥∃*𝑦 𝑥[Q]𝑦
28 dffun6 5804 . . . 4 (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
297, 27, 28mpbir2an 956 . . 3 Fun [Q]
30 dmss 5231 . . . . . 6 ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
313, 30ax-mp 5 . . . . 5 dom [Q] ⊆ dom ((N × N) × Q)
32 1nq 9605 . . . . . 6 1QQ
33 ne0i 3878 . . . . . 6 (1QQQ ≠ ∅)
34 dmxp 5251 . . . . . 6 (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
3532, 33, 34mp2b 10 . . . . 5 dom ((N × N) × Q) = (N × N)
3631, 35sseqtri 3598 . . . 4 dom [Q] ⊆ (N × N)
37 reurex 3135 . . . . . . . 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 7617 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
431breqi 4582 . . . . . . . . . . . 12 (𝑥[Q]𝑦𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44 brinxp2 5092 . . . . . . . . . . . 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 2998 . . . . . . . 8 (𝑥 ∈ (N × N) → (∃𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑥[Q]𝑦))
49 rexex 2983 . . . . . . . 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 3174 . . . . . . 7 𝑥 ∈ V
5352eldm 5229 . . . . . 6 (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
5451, 53sylibr 222 . . . . 5 (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
5554ssriv 3570 . . . 4 (N × N) ⊆ dom [Q]
5636, 55eqssi 3582 . . 3 dom [Q] = (N × N)
57 df-fn 5792 . . 3 ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
5829, 56, 57mpbir2an 956 . 2 [Q] Fn (N × N)
59 rnss 5261 . . . 4 ([Q] ⊆ ((N × N) × Q) → ran [Q] ⊆ ran ((N × N) × Q))
603, 59ax-mp 5 . . 3 ran [Q] ⊆ ran ((N × N) × Q)
61 rnxpss 5470 . . 3 ran ((N × N) × Q) ⊆ Q
6260, 61sstri 3575 . 2 ran [Q] ⊆ Q
63 df-f 5793 . 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 1976  ∃!weu 2456  ∃*wmo 2457  wne 2778  wrex 2895  ∃!wreu 2896  Vcvv 3171  cin 3537  wss 3538  c0 3872   class class class wbr 4576   × cxp 5025  dom cdm 5027  ran crn 5028  Rel wrel 5032  Fun wfun 5783   Fn wfn 5784  wf 5785   Er wer 7602  Ncnpi 9521   ~Q ceq 9528  Qcnq 9529  1Qc1q 9530  [Q]cerq 9531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2588  ax-sep 4702  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6823
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 1700  df-sb 1867  df-eu 2460  df-mo 2461  df-clab 2595  df-cleq 2601  df-clel 2604  df-nfc 2738  df-ne 2780  df-ral 2899  df-rex 2900  df-reu 2901  df-rmo 2902  df-rab 2903  df-v 3173  df-sbc 3401  df-csb 3498  df-dif 3541  df-un 3543  df-in 3545  df-ss 3552  df-pss 3554  df-nul 3873  df-if 4035  df-pw 4108  df-sn 4124  df-pr 4126  df-tp 4128  df-op 4130  df-uni 4366  df-iun 4450  df-br 4577  df-opab 4637  df-mpt 4638  df-tr 4674  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6934  df-1st 7035  df-2nd 7036  df-wrecs 7270  df-recs 7331  df-rdg 7369  df-1o 7423  df-oadd 7427  df-omul 7428  df-er 7605  df-ni 9549  df-mi 9551  df-lti 9552  df-enq 9588  df-nq 9589  df-erq 9590  df-1nq 9593
This theorem is referenced by:  nqercl  9608  nqerrel  9609  nqerid  9610  addnqf  9625  mulnqf  9626  adderpq  9633  mulerpq  9634  lterpq  9647
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