Theorem 2ndconst 6458

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 3831	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		fo2ndresm 6396	. . 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 3001	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝐴, 𝑦⟩
5	4	moani 2157	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)
6		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 5069	. . . . . . 7 ⊢ (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
8		fo2nd 6392	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 5617	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 5741	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦))
13	10, 11, 12	mp2an 430	. . . . . . . . 9 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦)
14	13	anbi1i 462	. . . . . . . 8 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
15		elxp7 6404	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)))
16		eleq1 2301	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑥) = 𝑦 → ((2nd ‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
17	16	biimpa 296	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) ∈ 𝐵) → 𝑦 ∈ 𝐵)
18	17	adantrl 482	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) → 𝑦 ∈ 𝐵)
19	18	adantrl 482	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑦 ∈ 𝐵)
20		elsni 3727	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥) ∈ {𝐴} → (1st ‘𝑥) = 𝐴)
21		eqopi 6406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝐴 ∧ (2nd ‘𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
22	21	ancom2s 572	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
23	22	an12s 571	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
24	20, 23	sylanr2 409	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
25	24	adantrrr 491	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
26	19, 25	jca 306	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
27	15, 26	sylan2b 287	. . . . . . . . . 10 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
28	27	adantl 277	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
29		fveq2 5695	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30		op2ndg 6385	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
31	6, 30	mpan2 429	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
32	29, 31	sylan9eqr 2293	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ⟨𝐴, 𝑦⟩) → (2nd ‘𝑥) = 𝑦)
33	32	adantrl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd ‘𝑥) = 𝑦)
34		simprr 537	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35		snidg 3738	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
36	35	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37		simprl 535	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦 ∈ 𝐵)
38		opelxpi 4806	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
39	36, 37, 38	syl2anc 415	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
40	34, 39	eqeltrd 2315	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
41	33, 40	jca 306	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
42	28, 41	impbida 604	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
43	14, 42	bitr3id 194	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
44	7, 43	bitrid 192	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
45	44	mobidv 2122	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
46	5, 45	mpbiri 168	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
47	46	alrimiv 1927	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48		funcnv2 5441	. . 3 ⊢ (Fun ◡(2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
49	47, 48	sylibr 134	. 2 ⊢ (𝐴 ∈ 𝑉 → Fun ◡(2nd ↾ ({𝐴} × 𝐵)))
50		dff1o3 5645	. 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 421	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821  {csn 3709  ⟨cop 3712   class class class wbr 4130   × cxp 4772  ^◡ccnv 4773   ↾ cres 4776  Fun wfun 5371   Fn wfn 5372  –onto→wfo 5375  –1-1-onto→wf1o 5376  ‘cfv 5377  1^st c1st 6372  2^nd c2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375

This theorem is used by:  xpfi  7239  fsum2dlemstep  12201  fprod2dlemstep  12389