Theorem 2ndconst 6112

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 3636	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		fo2ndresm 6053	. . 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 2854	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝐴, 𝑦⟩
5	4	moani 2067	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)
6		vex 2684	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 4820	. . . . . . 7 ⊢ (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
8		fo2nd 6049	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 5342	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 2684	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 5455	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦))
13	10, 11, 12	mp2an 422	. . . . . . . . 9 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦)
14	13	anbi1i 453	. . . . . . . 8 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
15		elxp7 6061	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)))
16		eleq1 2200	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑥) = 𝑦 → ((2nd ‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
17	16	biimpa 294	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) ∈ 𝐵) → 𝑦 ∈ 𝐵)
18	17	adantrl 469	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) → 𝑦 ∈ 𝐵)
19	18	adantrl 469	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑦 ∈ 𝐵)
20		elsni 3540	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥) ∈ {𝐴} → (1st ‘𝑥) = 𝐴)
21		eqopi 6063	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝐴 ∧ (2nd ‘𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
22	21	ancom2s 555	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
23	22	an12s 554	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
24	20, 23	sylanr2 402	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
25	24	adantrrr 478	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
26	19, 25	jca 304	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
27	15, 26	sylan2b 285	. . . . . . . . . 10 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
28	27	adantl 275	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
29		fveq2 5414	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30		op2ndg 6042	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
31	6, 30	mpan2 421	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
32	29, 31	sylan9eqr 2192	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ⟨𝐴, 𝑦⟩) → (2nd ‘𝑥) = 𝑦)
33	32	adantrl 469	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd ‘𝑥) = 𝑦)
34		simprr 521	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35		snidg 3549	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
36	35	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37		simprl 520	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦 ∈ 𝐵)
38		opelxpi 4566	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
39	36, 37, 38	syl2anc 408	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
40	34, 39	eqeltrd 2214	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
41	33, 40	jca 304	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
42	28, 41	impbida 585	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
43	14, 42	syl5bbr 193	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
44	7, 43	syl5bb 191	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
45	44	mobidv 2033	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
46	5, 45	mpbiri 167	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
47	46	alrimiv 1846	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48		funcnv2 5178	. . 3 ⊢ (Fun ◡(2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
49	47, 48	sylibr 133	. 2 ⊢ (𝐴 ∈ 𝑉 → Fun ◡(2nd ↾ ({𝐴} × 𝐵)))
50		dff1o3 5366	. 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 413	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃*wmo 1998  Vcvv 2681  {csn 3522  ⟨cop 3525   class class class wbr 3924   × cxp 4532  ^◡ccnv 4533   ↾ cres 4536  Fun wfun 5112   Fn wfn 5113  –onto→wfo 5116  –1-1-onto→wf1o 5117  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by:  xpfi  6811  fsum2dlemstep  11196