Theorem nqerf 11052

Description: Corollary of nqereu 11051: 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 11035	. . . . . . 7 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
2		inss2 4137	. . . . . . 7 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
3	1, 2	eqsstri 3959	. . . . . 6 ⊢ [Q] ⊆ ((N × N) × Q)
4		xpss 5672	. . . . . 6 ⊢ ((N × N) × Q) ⊆ (V × V)
5	3, 4	sstri 3922	. . . . 5 ⊢ [Q] ⊆ (V × V)
6		df-rel 5663	. . . . 5 ⊢ (Rel [Q] ↔ [Q] ⊆ (V × V))
7	5, 6	mpbir 234	. . . 4 ⊢ Rel [Q]
8		nqereu 11051	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → ∃!𝑦 ∈ Q 𝑦 ~Q 𝑥)
9		df-reu 3370	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
10		eumo 2606	. . . . . . . . 9 ⊢ (∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
11	9, 10	sylbi 220	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
12	8, 11	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
13		moanimv 2647	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
14	12, 13	mpbir 234	. . . . . 6 ⊢ ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
15	3	brel 5721	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q))
16	15	simpld 500	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑥 ∈ (N × N))
17	15	simprd 501	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ∈ Q)
18		enqer 11043	. . . . . . . . . 10 ⊢ ~Q Er (N × N)
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → ~Q Er (N × N))
20		inss1 4136	. . . . . . . . . . 11 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
21	1, 20	eqsstri 3959	. . . . . . . . . 10 ⊢ [Q] ⊆ ~Q
22	21	ssbri 5154	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → 𝑥 ~Q 𝑦)
23	19, 22	ersym 8766	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ~Q 𝑥)
24	16, 17, 23	jca32 525	. . . . . . 7 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
25	24	moimi 2573	. . . . . 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 1828	. . . 4 ⊢ ∀𝑥∃*𝑦 𝑥[Q]𝑦
28		dffun6 6575	. . . 4 ⊢ (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
29	7, 27, 28	mpbir2an 724	. . 3 ⊢ Fun [Q]
30		dmss 5894	. . . . . 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 11050	. . . . . 6 ⊢ 1Q ∈ Q
33		ne0i 4290	. . . . . 6 ⊢ (1Q ∈ Q → Q ≠ ∅)
34		dmxp 5921	. . . . . 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 3961	. . . 4 ⊢ dom [Q] ⊆ (N × N)
37		reurex 3373	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑦 ~Q 𝑥)
38		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39		simplr 781	. . . . . . . . . . 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 490	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
42	40, 41	ersym 8766	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
43	1	breqi 5112	. . . . . . . . . . . 12 ⊢ (𝑥[Q]𝑦 ↔ 𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44		brinxp2 5735	. . . . . . . . . . . 12 ⊢ (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
45	43, 44	bitri 278	. . . . . . . . . . 11 ⊢ (𝑥[Q]𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
46	38, 39, 42, 45	syl21anbrc 1363	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
47	46	ex 418	. . . . . . . . 9 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) → (𝑦 ~Q 𝑥 → 𝑥[Q]𝑦))
48	47	reximdva 3179	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → (∃𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑥[Q]𝑦))
49		rexex 3096	. . . . . . . 8 ⊢ (∃𝑦 ∈ Q 𝑥[Q]𝑦 → ∃𝑦 𝑥[Q]𝑦)
50	37, 48, 49	syl56 37	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 𝑥[Q]𝑦))
51	8, 50	mpd 16	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ∃𝑦 𝑥[Q]𝑦)
52		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
53	52	eldm 5892	. . . . . 6 ⊢ (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
54	51, 53	sylibr 237	. . . . 5 ⊢ (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
55	54	ssriv 3917	. . . 4 ⊢ (N × N) ⊆ dom [Q]
56	36, 55	eqssi 3929	. . 3 ⊢ dom [Q] = (N × N)
57		df-fn 6567	. . 3 ⊢ ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
58	29, 56, 57	mpbir2an 724	. 2 ⊢ [Q] Fn (N × N)
59	3	rnssi 5932	. . 3 ⊢ ran [Q] ⊆ ran ((N × N) × Q)
60		rnxpss 6180	. . 3 ⊢ ran ((N × N) × Q) ⊆ Q
61	59, 60	sstri 3922	. 2 ⊢ ran [Q] ⊆ Q
62		df-f 6568	. 2 ⊢ ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
63	58, 61, 62	mpbir2an 724	1 ⊢ [Q]:(N × N)⟶Q

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃*wmo 2565  ∃!weu 2596   ≠ wne 2959  ∃wrex 3090  ∃!wreu 3367  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   class class class wbr 5106   × cxp 5654  dom cdm 5656  ran crn 5657  Rel wrel 5661  Fun wfun 6558   Fn wfn 6559  ⟶wf 6560   Er wer 8750  Ncnpi 10966   ~_Q ceq 10973  Qcnq 10974  1_Qc1q 10975  [Q]cerq 10976

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-oadd 8516  df-omul 8517  df-er 8753  df-ni 10994  df-mi 10996  df-lti 10997  df-enq 11033  df-nq 11034  df-erq 11035  df-1nq 11038

This theorem is used by:  nqercl  11053  nqerrel  11054  nqerid  11055  addnqf  11070  mulnqf  11071  adderpq  11078  mulerpq  11079  lterpq  11092