Theorem nqerf 10832

Description: Corollary of nqereu 10831: 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.

Step	Hyp	Ref	Expression
1		df-erq 10815	. . . . . . 7 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
2		inss2 4187	. . . . . . 7 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
3	1, 2	eqsstri 3977	. . . . . 6 ⊢ [Q] ⊆ ((N × N) × Q)
4		xpss 5637	. . . . . 6 ⊢ ((N × N) × Q) ⊆ (V × V)
5	3, 4	sstri 3940	. . . . 5 ⊢ [Q] ⊆ (V × V)
6		df-rel 5628	. . . . 5 ⊢ (Rel [Q] ↔ [Q] ⊆ (V × V))
7	5, 6	mpbir 231	. . . 4 ⊢ Rel [Q]
8		nqereu 10831	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → ∃!𝑦 ∈ Q 𝑦 ~Q 𝑥)
9		df-reu 3348	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
10		eumo 2575	. . . . . . . . 9 ⊢ (∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
11	9, 10	sylbi 217	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
12	8, 11	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
13		moanimv 2616	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
14	12, 13	mpbir 231	. . . . . 6 ⊢ ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
15	3	brel 5686	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q))
16	15	simpld 494	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑥 ∈ (N × N))
17	15	simprd 495	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ∈ Q)
18		enqer 10823	. . . . . . . . . 10 ⊢ ~Q Er (N × N)
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → ~Q Er (N × N))
20		inss1 4186	. . . . . . . . . . 11 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
21	1, 20	eqsstri 3977	. . . . . . . . . 10 ⊢ [Q] ⊆ ~Q
22	21	ssbri 5140	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → 𝑥 ~Q 𝑦)
23	19, 22	ersym 8643	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ~Q 𝑥)
24	16, 17, 23	jca32 515	. . . . . . 7 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
25	24	moimi 2542	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)) → ∃*𝑦 𝑥[Q]𝑦)
26	14, 25	ax-mp 5	. . . . 5 ⊢ ∃*𝑦 𝑥[Q]𝑦
27	26	ax-gen 1796	. . . 4 ⊢ ∀𝑥∃*𝑦 𝑥[Q]𝑦
28		dffun6 6500	. . . 4 ⊢ (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
29	7, 27, 28	mpbir2an 711	. . 3 ⊢ Fun [Q]
30		dmss 5848	. . . . . 6 ⊢ ([Q] ⊆ ((N × N) × Q) → dom [Q] ⊆ dom ((N × N) × Q))
31	3, 30	ax-mp 5	. . . . 5 ⊢ dom [Q] ⊆ dom ((N × N) × Q)
32		1nq 10830	. . . . . 6 ⊢ 1Q ∈ Q
33		ne0i 4290	. . . . . 6 ⊢ (1Q ∈ Q → Q ≠ ∅)
34		dmxp 5875	. . . . . 6 ⊢ (Q ≠ ∅ → dom ((N × N) × Q) = (N × N))
35	32, 33, 34	mp2b 10	. . . . 5 ⊢ dom ((N × N) × Q) = (N × N)
36	31, 35	sseqtri 3979	. . . 4 ⊢ dom [Q] ⊆ (N × N)
37		reurex 3351	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑦 ~Q 𝑥)
38		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39		simplr 768	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ∈ Q)
40	18	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → ~Q Er (N × N))
41		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
42	40, 41	ersym 8643	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
43	1	breqi 5101	. . . . . . . . . . . 12 ⊢ (𝑥[Q]𝑦 ↔ 𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44		brinxp2 5699	. . . . . . . . . . . 12 ⊢ (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
45	43, 44	bitri 275	. . . . . . . . . . 11 ⊢ (𝑥[Q]𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
46	38, 39, 42, 45	syl21anbrc 1345	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
47	46	ex 412	. . . . . . . . 9 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) → (𝑦 ~Q 𝑥 → 𝑥[Q]𝑦))
48	47	reximdva 3146	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → (∃𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑥[Q]𝑦))
49		rexex 3063	. . . . . . . 8 ⊢ (∃𝑦 ∈ Q 𝑥[Q]𝑦 → ∃𝑦 𝑥[Q]𝑦)
50	37, 48, 49	syl56 36	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 𝑥[Q]𝑦))
51	8, 50	mpd 15	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ∃𝑦 𝑥[Q]𝑦)
52		vex 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
53	52	eldm 5846	. . . . . 6 ⊢ (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
54	51, 53	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
55	54	ssriv 3934	. . . 4 ⊢ (N × N) ⊆ dom [Q]
56	36, 55	eqssi 3947	. . 3 ⊢ dom [Q] = (N × N)
57		df-fn 6492	. . 3 ⊢ ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
58	29, 56, 57	mpbir2an 711	. 2 ⊢ [Q] Fn (N × N)
59	3	rnssi 5886	. . 3 ⊢ ran [Q] ⊆ ran ((N × N) × Q)
60		rnxpss 6127	. . 3 ⊢ ran ((N × N) × Q) ⊆ Q
61	59, 60	sstri 3940	. 2 ⊢ ran [Q] ⊆ Q
62		df-f 6493	. 2 ⊢ ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
63	58, 61, 62	mpbir2an 711	1 ⊢ [Q]:(N × N)⟶Q

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  ∃!weu 2565   ≠ wne 2929  ∃wrex 3057  ∃!wreu 3345  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282   class class class wbr 5095   × cxp 5619  dom cdm 5621  ran crn 5622  Rel wrel 5626  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485   Er wer 8628  Ncnpi 10746   ~_Q ceq 10753  Qcnq 10754  1_Qc1q 10755  [Q]cerq 10756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-omul 8399  df-er 8631  df-ni 10774  df-mi 10776  df-lti 10777  df-enq 10813  df-nq 10814  df-erq 10815  df-1nq 10818

This theorem is referenced by:  nqercl  10833  nqerrel  10834  nqerid  10835  addnqf  10850  mulnqf  10851  adderpq  10858  mulerpq  10859  lterpq  10872