Theorem nqerf 10844

Description: Corollary of nqereu 10843: 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 10827	. . . . . . 7 ⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
2		inss2 4179	. . . . . . 7 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ((N × N) × Q)
3	1, 2	eqsstri 3969	. . . . . 6 ⊢ [Q] ⊆ ((N × N) × Q)
4		xpss 5640	. . . . . 6 ⊢ ((N × N) × Q) ⊆ (V × V)
5	3, 4	sstri 3932	. . . . 5 ⊢ [Q] ⊆ (V × V)
6		df-rel 5631	. . . . 5 ⊢ (Rel [Q] ↔ [Q] ⊆ (V × V))
7	5, 6	mpbir 231	. . . 4 ⊢ Rel [Q]
8		nqereu 10843	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → ∃!𝑦 ∈ Q 𝑦 ~Q 𝑥)
9		df-reu 3344	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 ↔ ∃!𝑦(𝑦 ∈ Q ∧ 𝑦 ~Q 𝑥))
10		eumo 2579	. . . . . . . . 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 2620	. . . . . . 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 5689	. . . . . . . . 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 10835	. . . . . . . . . 10 ⊢ ~Q Er (N × N)
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → ~Q Er (N × N))
20		inss1 4178	. . . . . . . . . . 11 ⊢ ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
21	1, 20	eqsstri 3969	. . . . . . . . . 10 ⊢ [Q] ⊆ ~Q
22	21	ssbri 5131	. . . . . . . . 9 ⊢ (𝑥[Q]𝑦 → 𝑥 ~Q 𝑦)
23	19, 22	ersym 8649	. . . . . . . 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 2546	. . . . . 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 1797	. . . 4 ⊢ ∀𝑥∃*𝑦 𝑥[Q]𝑦
28		dffun6 6503	. . . 4 ⊢ (Fun [Q] ↔ (Rel [Q] ∧ ∀𝑥∃*𝑦 𝑥[Q]𝑦))
29	7, 27, 28	mpbir2an 712	. . 3 ⊢ Fun [Q]
30		dmss 5851	. . . . . 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 10842	. . . . . 6 ⊢ 1Q ∈ Q
33		ne0i 4282	. . . . . 6 ⊢ (1Q ∈ Q → Q ≠ ∅)
34		dmxp 5878	. . . . . 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 3971	. . . 4 ⊢ dom [Q] ⊆ (N × N)
37		reurex 3347	. . . . . . . 8 ⊢ (∃!𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑦 ~Q 𝑥)
38		simpll 767	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ∈ (N × N))
39		simplr 769	. . . . . . . . . . 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 8649	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ Q) ∧ 𝑦 ~Q 𝑥) → 𝑥 ~Q 𝑦)
43	1	breqi 5092	. . . . . . . . . . . 12 ⊢ (𝑥[Q]𝑦 ↔ 𝑥( ~Q ∩ ((N × N) × Q))𝑦)
44		brinxp2 5702	. . . . . . . . . . . 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 1346	. . . . . . . . . 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 3151	. . . . . . . 8 ⊢ (𝑥 ∈ (N × N) → (∃𝑦 ∈ Q 𝑦 ~Q 𝑥 → ∃𝑦 ∈ Q 𝑥[Q]𝑦))
49		rexex 3068	. . . . . . . 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 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
53	52	eldm 5849	. . . . . 6 ⊢ (𝑥 ∈ dom [Q] ↔ ∃𝑦 𝑥[Q]𝑦)
54	51, 53	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ (N × N) → 𝑥 ∈ dom [Q])
55	54	ssriv 3926	. . . 4 ⊢ (N × N) ⊆ dom [Q]
56	36, 55	eqssi 3939	. . 3 ⊢ dom [Q] = (N × N)
57		df-fn 6495	. . 3 ⊢ ([Q] Fn (N × N) ↔ (Fun [Q] ∧ dom [Q] = (N × N)))
58	29, 56, 57	mpbir2an 712	. 2 ⊢ [Q] Fn (N × N)
59	3	rnssi 5889	. . 3 ⊢ ran [Q] ⊆ ran ((N × N) × Q)
60		rnxpss 6130	. . 3 ⊢ ran ((N × N) × Q) ⊆ Q
61	59, 60	sstri 3932	. 2 ⊢ ran [Q] ⊆ Q
62		df-f 6496	. 2 ⊢ ([Q]:(N × N)⟶Q ↔ ([Q] Fn (N × N) ∧ ran [Q] ⊆ Q))
63	58, 61, 62	mpbir2an 712	1 ⊢ [Q]:(N × N)⟶Q

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569   ≠ wne 2933  ∃wrex 3062  ∃!wreu 3341  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086   × cxp 5622  dom cdm 5624  ran crn 5625  Rel wrel 5629  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488   Er wer 8633  Ncnpi 10758   ~_Q ceq 10765  Qcnq 10766  1_Qc1q 10767  [Q]cerq 10768

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-omul 8403  df-er 8636  df-ni 10786  df-mi 10788  df-lti 10789  df-enq 10825  df-nq 10826  df-erq 10827  df-1nq 10830

This theorem is referenced by:  nqercl  10845  nqerrel  10846  nqerid  10847  addnqf  10862  mulnqf  10863  adderpq  10870  mulerpq  10871  lterpq  10884