Theorem 2ndconst 6331

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 3761	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		fo2ndresm 6271	. . 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 2955	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝐴, 𝑦⟩
5	4	moani 2126	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)
6		vex 2779	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 4984	. . . . . . 7 ⊢ (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
8		fo2nd 6267	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 5522	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 2779	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 5642	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦))
13	10, 11, 12	mp2an 426	. . . . . . . . 9 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦)
14	13	anbi1i 458	. . . . . . . 8 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
15		elxp7 6279	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)))
16		eleq1 2270	. . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) → 𝑦 ∈ 𝐵)
19	18	adantrl 478	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑦 ∈ 𝐵)
20		elsni 3661	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥) ∈ {𝐴} → (1st ‘𝑥) = 𝐴)
21		eqopi 6281	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝐴 ∧ (2nd ‘𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
22	21	ancom2s 566	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
23	22	an12s 565	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
24	20, 23	sylanr2 405	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
25	24	adantrrr 487	. . . . . . . . . . . 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 5599	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30		op2ndg 6260	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
31	6, 30	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
32	29, 31	sylan9eqr 2262	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ⟨𝐴, 𝑦⟩) → (2nd ‘𝑥) = 𝑦)
33	32	adantrl 478	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd ‘𝑥) = 𝑦)
34		simprr 531	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35		snidg 3672	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
36	35	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦 ∈ 𝐵)
38		opelxpi 4725	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
39	36, 37, 38	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
40	34, 39	eqeltrd 2284	. . . . . . . . . 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 596	. . . . . . . 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 2091	. . . . 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 1898	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48		funcnv2 5353	. . 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 5550	. 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 417	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516  ∃*wmo 2056   ∈ wcel 2178  Vcvv 2776  {csn 3643  ⟨cop 3646   class class class wbr 4059   × cxp 4691  ^◡ccnv 4692   ↾ cres 4695  Fun wfun 5284   Fn wfn 5285  –onto→wfo 5288  –1-1-onto→wf1o 5289  ‘cfv 5290  1^st c1st 6247  2^nd c2nd 6248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250

This theorem is referenced by:  xpfi  7055  fsum2dlemstep  11860  fprod2dlemstep  12048