Theorem 2ndconst 5782

Description: The mapping of a restriction of the 2^nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Assertion

Ref	Expression
2ndconst	⊢ (A ∈ 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–1-1-onto→B)

Proof of Theorem 2ndconst

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

Step	Hyp	Ref	Expression
1		snmg 3476	. . 3 ⊢ (A ∈ 𝑉 → ∃x x ∈ {A})
2		fo2ndresm 5728	. . 3 ⊢ (∃x x ∈ {A} → (2nd ↾ ({A} × B)):({A} × B)–onto→B)
3	1, 2	syl 14	. 2 ⊢ (A ∈ 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–onto→B)
4		moeq 2710	. . . . . 6 ⊢ ∃*x x = ⟨A, y⟩
5	4	moani 1967	. . . . 5 ⊢ ∃*x(y ∈ B ∧ x = ⟨A, y⟩)
6		vex 2554	. . . . . . . 8 ⊢ y ∈ V
7	6	brres 4560	. . . . . . 7 ⊢ (x(2nd ↾ ({A} × B))y ↔ (x2nd y ∧ x ∈ ({A} × B)))
8		fo2nd 5724	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 5049	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 7	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 2554	. . . . . . . . . 10 ⊢ x ∈ V
12		fnbrfvb 5155	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ x ∈ V) → ((2nd ‘x) = y ↔ x2nd y))
13	10, 11, 12	mp2an 402	. . . . . . . . 9 ⊢ ((2nd ‘x) = y ↔ x2nd y)
14	13	anbi1i 431	. . . . . . . 8 ⊢ (((2nd ‘x) = y ∧ x ∈ ({A} × B)) ↔ (x2nd y ∧ x ∈ ({A} × B)))
15		elxp7 5736	. . . . . . . . . . 11 ⊢ (x ∈ ({A} × B) ↔ (x ∈ (V × V) ∧ ((1st ‘x) ∈ {A} ∧ (2nd ‘x) ∈ B)))
16		eleq1 2097	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘x) = y → ((2nd ‘x) ∈ B ↔ y ∈ B))
17	16	biimpa 280	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘x) = y ∧ (2nd ‘x) ∈ B) → y ∈ B)
18	17	adantrl 447	. . . . . . . . . . . . 13 ⊢ (((2nd ‘x) = y ∧ ((1st ‘x) ∈ {A} ∧ (2nd ‘x) ∈ B)) → y ∈ B)
19	18	adantrl 447	. . . . . . . . . . . 12 ⊢ (((2nd ‘x) = y ∧ (x ∈ (V × V) ∧ ((1st ‘x) ∈ {A} ∧ (2nd ‘x) ∈ B))) → y ∈ B)
20		elsni 3390	. . . . . . . . . . . . . 14 ⊢ ((1st ‘x) ∈ {A} → (1st ‘x) = A)
21		eqopi 5737	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ (V × V) ∧ ((1st ‘x) = A ∧ (2nd ‘x) = y)) → x = ⟨A, y⟩)
22	21	ancom2s 500	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ (V × V) ∧ ((2nd ‘x) = y ∧ (1st ‘x) = A)) → x = ⟨A, y⟩)
23	22	an12s 499	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘x) = y ∧ (x ∈ (V × V) ∧ (1st ‘x) = A)) → x = ⟨A, y⟩)
24	20, 23	sylanr2 385	. . . . . . . . . . . . 13 ⊢ (((2nd ‘x) = y ∧ (x ∈ (V × V) ∧ (1st ‘x) ∈ {A})) → x = ⟨A, y⟩)
25	24	adantrrr 456	. . . . . . . . . . . 12 ⊢ (((2nd ‘x) = y ∧ (x ∈ (V × V) ∧ ((1st ‘x) ∈ {A} ∧ (2nd ‘x) ∈ B))) → x = ⟨A, y⟩)
26	19, 25	jca 290	. . . . . . . . . . 11 ⊢ (((2nd ‘x) = y ∧ (x ∈ (V × V) ∧ ((1st ‘x) ∈ {A} ∧ (2nd ‘x) ∈ B))) → (y ∈ B ∧ x = ⟨A, y⟩))
27	15, 26	sylan2b 271	. . . . . . . . . 10 ⊢ (((2nd ‘x) = y ∧ x ∈ ({A} × B)) → (y ∈ B ∧ x = ⟨A, y⟩))
28	27	adantl 262	. . . . . . . . 9 ⊢ ((A ∈ 𝑉 ∧ ((2nd ‘x) = y ∧ x ∈ ({A} × B))) → (y ∈ B ∧ x = ⟨A, y⟩))
29		fveq2 5119	. . . . . . . . . . . 12 ⊢ (x = ⟨A, y⟩ → (2nd ‘x) = (2nd ‘⟨A, y⟩))
30		op2ndg 5717	. . . . . . . . . . . . 13 ⊢ ((A ∈ 𝑉 ∧ y ∈ V) → (2nd ‘⟨A, y⟩) = y)
31	6, 30	mpan2 401	. . . . . . . . . . . 12 ⊢ (A ∈ 𝑉 → (2nd ‘⟨A, y⟩) = y)
32	29, 31	sylan9eqr 2091	. . . . . . . . . . 11 ⊢ ((A ∈ 𝑉 ∧ x = ⟨A, y⟩) → (2nd ‘x) = y)
33	32	adantrl 447	. . . . . . . . . 10 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → (2nd ‘x) = y)
34		simprr 484	. . . . . . . . . . 11 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → x = ⟨A, y⟩)
35		snidg 3391	. . . . . . . . . . . . 13 ⊢ (A ∈ 𝑉 → A ∈ {A})
36	35	adantr 261	. . . . . . . . . . . 12 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → A ∈ {A})
37		simprl 483	. . . . . . . . . . . 12 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → y ∈ B)
38		opelxpi 4318	. . . . . . . . . . . 12 ⊢ ((A ∈ {A} ∧ y ∈ B) → ⟨A, y⟩ ∈ ({A} × B))
39	36, 37, 38	syl2anc 391	. . . . . . . . . . 11 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → ⟨A, y⟩ ∈ ({A} × B))
40	34, 39	eqeltrd 2111	. . . . . . . . . 10 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → x ∈ ({A} × B))
41	33, 40	jca 290	. . . . . . . . 9 ⊢ ((A ∈ 𝑉 ∧ (y ∈ B ∧ x = ⟨A, y⟩)) → ((2nd ‘x) = y ∧ x ∈ ({A} × B)))
42	28, 41	impbida 528	. . . . . . . 8 ⊢ (A ∈ 𝑉 → (((2nd ‘x) = y ∧ x ∈ ({A} × B)) ↔ (y ∈ B ∧ x = ⟨A, y⟩)))
43	14, 42	syl5bbr 183	. . . . . . 7 ⊢ (A ∈ 𝑉 → ((x2nd y ∧ x ∈ ({A} × B)) ↔ (y ∈ B ∧ x = ⟨A, y⟩)))
44	7, 43	syl5bb 181	. . . . . 6 ⊢ (A ∈ 𝑉 → (x(2nd ↾ ({A} × B))y ↔ (y ∈ B ∧ x = ⟨A, y⟩)))
45	44	mobidv 1933	. . . . 5 ⊢ (A ∈ 𝑉 → (∃*x x(2nd ↾ ({A} × B))y ↔ ∃*x(y ∈ B ∧ x = ⟨A, y⟩)))
46	5, 45	mpbiri 157	. . . 4 ⊢ (A ∈ 𝑉 → ∃*x x(2nd ↾ ({A} × B))y)
47	46	alrimiv 1751	. . 3 ⊢ (A ∈ 𝑉 → ∀y∃*x x(2nd ↾ ({A} × B))y)
48		funcnv2 4900	. . 3 ⊢ (Fun ◡(2nd ↾ ({A} × B)) ↔ ∀y∃*x x(2nd ↾ ({A} × B))y)
49	47, 48	sylibr 137	. 2 ⊢ (A ∈ 𝑉 → Fun ◡(2nd ↾ ({A} × B)))
50		dff1o3 5073	. 2 ⊢ ((2nd ↾ ({A} × B)):({A} × B)–1-1-onto→B ↔ ((2nd ↾ ({A} × B)):({A} × B)–onto→B ∧ Fun ◡(2nd ↾ ({A} × B))))
51	3, 49, 50	sylanbrc 394	1 ⊢ (A ∈ 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–1-1-onto→B)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  Vcvv 2551  {csn 3366  ⟨cop 3369   class class class wbr 3754   × cxp 4285  ^◡ccnv 4286   ↾ cres 4289  Fun wfun 4838   Fn wfn 4839  –onto→wfo 4842  –1-1-onto→wf1o 4843  ‘cfv 4844  1^st c1st 5704  2^nd c2nd 5705

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-1st 5706  df-2nd 5707

This theorem is referenced by: (None)