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Theorem nqerf 10866
Description: Corollary of nqereu 10865: 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 10849 . . . . . . 7 [Q] = ( ~Q ∩ ((N × N) × Q))
2 inss2 4189 . . . . . . 7 ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
31, 2eqsstri 3978 . . . . . 6 [Q] ⊆ ((N × N) × Q)
4 xpss 5649 . . . . . 6 ((N × N) × Q) ⊆ (V × V)
53, 4sstri 3953 . . . . 5 [Q] ⊆ (V × V)
6 df-rel 5640 . . . . 5 (Rel [Q] ↔ [Q] ⊆ (V × V))
75, 6mpbir 230 . . . 4 Rel [Q]
8 nqereu 10865 . . . . . . . 8 (𝑥 ∈ (N × N) → ∃!𝑦Q 𝑦 ~Q 𝑥)
9 df-reu 3354 . . . . . . . . 9 (∃!𝑦Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦Q𝑦 ~Q 𝑥))
10 eumo 2576 . . . . . . . . 9 (∃!𝑦(𝑦Q𝑦 ~Q 𝑥) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
119, 10sylbi 216 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
128, 11syl 17 . . . . . . 7 (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥))
13 moanimv 2619 . . . . . . 7 (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦Q𝑦 ~Q 𝑥)))
1412, 13mpbir 230 . . . . . 6 ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥))
153brel 5697 . . . . . . . . 9 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦Q))
1615simpld 495 . . . . . . . 8 (𝑥[Q]𝑦𝑥 ∈ (N × N))
1715simprd 496 . . . . . . . 8 (𝑥[Q]𝑦𝑦Q)
18 enqer 10857 . . . . . . . . . 10 ~Q Er (N × N)
1918a1i 11 . . . . . . . . 9 (𝑥[Q]𝑦 → ~Q Er (N × N))
20 inss1 4188 . . . . . . . . . . 11 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
211, 20eqsstri 3978 . . . . . . . . . 10 [Q] ⊆ ~Q
2221ssbri 5150 . . . . . . . . 9 (𝑥[Q]𝑦𝑥 ~Q 𝑦)
2319, 22ersym 8660 . . . . . . . 8 (𝑥[Q]𝑦𝑦 ~Q 𝑥)
2416, 17, 23jca32 516 . . . . . . 7 (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦Q𝑦 ~Q 𝑥)))
2524moimi 2543 . . . . . 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 6509 . . . 4 (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
297, 27, 28mpbir2an 709 . . 3 Fun [Q]
30 dmss 5858 . . . . . 6 ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
313, 30ax-mp 5 . . . . 5 dom [Q] ⊆ dom ((N × N) × Q)
32 1nq 10864 . . . . . 6 1QQ
33 ne0i 4294 . . . . . 6 (1QQQ ≠ ∅)
34 dmxp 5884 . . . . . 6 (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
3532, 33, 34mp2b 10 . . . . 5 dom ((N × N) × Q) = (N × N)
3631, 35sseqtri 3980 . . . 4 dom [Q] ⊆ (N × N)
37 reurex 3357 . . . . . . . 8 (∃!𝑦Q 𝑦 ~Q 𝑥 → ∃𝑦Q 𝑦 ~Q 𝑥)
38 simpll 765 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39 simplr 767 . . . . . . . . . . 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 485 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
4240, 41ersym 8660 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
431breqi 5111 . . . . . . . . . . . 12 (𝑥[Q]𝑦𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44 brinxp2 5709 . . . . . . . . . . . 12 (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑥 ~Q 𝑦))
4543, 44bitri 274 . . . . . . . . . . 11 (𝑥[Q]𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑥 ~Q 𝑦))
4638, 39, 42, 45syl21anbrc 1344 . . . . . . . . . 10 (((𝑥 ∈ (N × N) ∧ 𝑦Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
4746ex 413 . . . . . . . . 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 3079 . . . . . . . 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 3449 . . . . . . 7 𝑥 ∈ V
5352eldm 5856 . . . . . 6 (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
5451, 53sylibr 233 . . . . 5 (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
5554ssriv 3948 . . . 4 (N × N) ⊆ dom [Q]
5636, 55eqssi 3960 . . 3 dom [Q] = (N × N)
57 df-fn 6499 . . 3 ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
5829, 56, 57mpbir2an 709 . 2 [Q] Fn (N × N)
593rnssi 5895 . . 3 ran [Q] ⊆ ran ((N × N) × Q)
60 rnxpss 6124 . . 3 ran ((N × N) × Q) ⊆ Q
6159, 60sstri 3953 . 2 ran [Q] ⊆ Q
62 df-f 6500 . 2 ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
6358, 61, 62mpbir2an 709 1 [Q]:(N × N)⟶Q
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃*wmo 2536  ∃!weu 2566  wne 2943  wrex 3073  ∃!wreu 3351  Vcvv 3445  cin 3909  wss 3910  c0 4282   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  Rel wrel 5638  Fun wfun 6490   Fn wfn 6491  wf 6492   Er wer 8645  Ncnpi 10780   ~Q ceq 10787  Qcnq 10788  1Qc1q 10789  [Q]cerq 10790
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ni 10808  df-mi 10810  df-lti 10811  df-enq 10847  df-nq 10848  df-erq 10849  df-1nq 10852
This theorem is referenced by:  nqercl  10867  nqerrel  10868  nqerid  10869  addnqf  10884  mulnqf  10885  adderpq  10892  mulerpq  10893  lterpq  10906
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