Theorem 1stconst 6126

Description: The mapping of a restriction of the 1^st function to a constant function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
1stconst	⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Proof of Theorem 1stconst

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snmg 3649	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐵})
2		fo1stresm 6067	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐵} → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
4		moeq 2863	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2070	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 2692	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 4833	. . . . . . 7 ⊢ (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
8		fo1st 6063	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
9		fofn 5355	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
11		vex 2692	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 5470	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ 𝑥 ∈ V) → ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦))
13	10, 11, 12	mp2an 423	. . . . . . . . 9 ⊢ ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦)
14	13	anbi1i 454	. . . . . . . 8 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
15		elxp7 6076	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2203	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = 𝑦 → ((1st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpa 294	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) ∈ 𝐴) → 𝑦 ∈ 𝐴)
18	17	adantrr 471	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑦 ∈ 𝐴)
19	18	adantrl 470	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑦 ∈ 𝐴)
20		elsni 3550	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑥) ∈ {𝐵} → (2nd ‘𝑥) = 𝐵)
21		eqopi 6078	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	an12s 555	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanr2 403	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantrrl 478	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 304	. . . . . . . . . . 11 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylan2b 285	. . . . . . . . . 10 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 275	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 522	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29	28	fveq2d 5433	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30		simprl 521	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
31		simpl 108	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
32		op1stg 6056	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
33	30, 31, 32	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
34	29, 33	eqtrd 2173	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = 𝑦)
35		snidg 3561	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
36	35	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37		opelxpi 4579	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
38	30, 36, 37	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
39	28, 38	eqeltrd 2217	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
40	34, 39	jca 304	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
41	27, 40	impbida 586	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	14, 41	bitr3id 193	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	7, 42	syl5bb 191	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	43	mobidv 2036	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
45	5, 44	mpbiri 167	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46	45	alrimiv 1847	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47		funcnv2 5191	. . 3 ⊢ (Fun ◡(1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
48	46, 47	sylibr 133	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ◡(1st ↾ (𝐴 × {𝐵})))
49		dff1o3 5381	. 2 ⊢ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴 ∧ Fun ◡(1st ↾ (𝐴 × {𝐵}))))
50	3, 48, 49	sylanbrc 414	1 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃*wmo 2001  Vcvv 2689  {csn 3532  ⟨cop 3535   class class class wbr 3937   × cxp 4545  ^◡ccnv 4546   ↾ cres 4549  Fun wfun 5125   Fn wfn 5126  –onto→wfo 5129  –1-1-onto→wf1o 5130  ‘cfv 5131  1^st c1st 6044  2^nd c2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by: (None)