Theorem nqerf 10089

Description: Corollary of nqereu 10088: 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 10072	. . . . . . 7 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
2		inss2 4054	. . . . . . 7 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
3	1, 2	eqsstri 3854	. . . . . 6 ⊢ [Q] ⊆ ((N × N) × Q)
4		xpss 5373	. . . . . 6 ⊢ ((N × N) × Q) ⊆ (V × V)
5	3, 4	sstri 3830	. . . . 5 ⊢ [Q] ⊆ (V × V)
6		df-rel 5364	. . . . 5 ⊢ (Rel [Q] ↔ [Q] ⊆ (V × V))
7	5, 6	mpbir 223	. . . 4 ⊢ Rel [Q]
8		nqereu 10088	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → ∃!𝑦 ∈ Q 𝑦 ~Q 𝑥)
9		df-reu 3097	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
10		eumo 2598	. . . . . . . . 9 ⊢ (∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
11	9, 10	sylbi 209	. . . . . . . 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 2654	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)) ↔ (𝑥 ∈ (N × N) → ∃*𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
14	12, 13	mpbir 223	. . . . . 6 ⊢ ∃*𝑦(𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
15	3	brel 5416	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q))
16	15	simpld 490	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑥 ∈ (N × N))
17	15	simprd 491	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ∈ Q)
18		enqer 10080	. . . . . . . . . 10 ⊢ ~Q Er (N × N)
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → ~Q Er (N × N))
20		inss1 4053	. . . . . . . . . . 11 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
21	1, 20	eqsstri 3854	. . . . . . . . . 10 ⊢ [Q] ⊆ ~Q
22	21	ssbri 4933	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → 𝑥 ~Q 𝑦)
23	19, 22	ersym 8040	. . . . . . . 8 ⊢ (𝑥[Q]𝑦 → 𝑦 ~Q 𝑥)
24	16, 17, 23	jca32 511	. . . . . . 7 ⊢ (𝑥[Q]𝑦 → (𝑥 ∈ (N × N) ∧ (𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥)))
25	24	moimi 2557	. . . . . 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 1839	. . . 4 ⊢ ∀𝑥∃*𝑦 𝑥[Q]𝑦
28		dffun6 6152	. . . 4 ⊢ (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
29	7, 27, 28	mpbir2an 701	. . 3 ⊢ Fun [Q]
30		dmss 5570	. . . . . 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 10087	. . . . . 6 ⊢ 1Q ∈ Q
33		ne0i 4149	. . . . . 6 ⊢ (1Q ∈ Q → Q ≠ ∅)
34		dmxp 5591	. . . . . 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 3856	. . . 4 ⊢ dom [Q] ⊆ (N × N)
37		reurex 3356	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑦 ~Q 𝑥)
38		simpll 757	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39		simplr 759	. . . . . . . . . . 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 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑦 ~Q 𝑥)
42	40, 41	ersym 8040	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
43	1	breqi 4894	. . . . . . . . . . . 12 ⊢ (𝑥[Q]𝑦 ↔ 𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44		brinxp2 5428	. . . . . . . . . . . 12 ⊢ (𝑥( ~Q ∩ ((N × N) × Q))𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
45	43, 44	bitri 267	. . . . . . . . . . 11 ⊢ (𝑥[Q]𝑦 ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑥 ~Q 𝑦))
46	38, 39, 42, 45	syl21anbrc 1401	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥[Q]𝑦)
47	46	ex 403	. . . . . . . . 9 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) → (𝑦 ~Q 𝑥 → 𝑥[Q]𝑦))
48	47	reximdva 3198	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → (∃𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑥[Q]𝑦))
49		rexex 3183	. . . . . . . 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 3401	. . . . . . 7 ⊢ 𝑥 ∈ V
53	52	eldm 5568	. . . . . 6 ⊢ (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
54	51, 53	sylibr 226	. . . . 5 ⊢ (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
55	54	ssriv 3825	. . . 4 ⊢ (N × N) ⊆ dom [Q]
56	36, 55	eqssi 3837	. . 3 ⊢ dom [Q] = (N × N)
57		df-fn 6140	. . 3 ⊢ ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
58	29, 56, 57	mpbir2an 701	. 2 ⊢ [Q] Fn (N × N)
59	3	rnssi 5602	. . 3 ⊢ ran [Q] ⊆ ran ((N × N) × Q)
60		rnxpss 5822	. . 3 ⊢ ran ((N × N) × Q) ⊆ Q
61	59, 60	sstri 3830	. 2 ⊢ ran [Q] ⊆ Q
62		df-f 6141	. 2 ⊢ ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
63	58, 61, 62	mpbir2an 701	1 ⊢ [Q]:(N × N)⟶Q

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∃*wmo 2549  ∃!weu 2586   ≠ wne 2969  ∃wrex 3091  ∃!wreu 3092  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141   class class class wbr 4888   × cxp 5355  dom cdm 5357  ran crn 5358  Rel wrel 5362  Fun wfun 6131   Fn wfn 6132  ⟶wf 6133   Er wer 8025  Ncnpi 10003   ~_Q ceq 10010  Qcnq 10011  1_Qc1q 10012  [Q]cerq 10013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850  df-er 8028  df-ni 10031  df-mi 10033  df-lti 10034  df-enq 10070  df-nq 10071  df-erq 10072  df-1nq 10075

This theorem is referenced by:  nqercl  10090  nqerrel  10091  nqerid  10092  addnqf  10107  mulnqf  10108  adderpq  10115  mulerpq  10116  lterpq  10129