ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2ndconst GIF version

Theorem 2ndconst 6025
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
2ndconst (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)

Proof of Theorem 2ndconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3580 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 fo2ndresm 5971 . . 3 (∃𝑥 𝑥 ∈ {𝐴} → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 14 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 2804 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2025 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 2636 . . . . . . . 8 𝑦 ∈ V
76brres 4751 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
8 fo2nd 5967 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5270 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 7 . . . . . . . . . 10 2nd Fn V
11 vex 2636 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5380 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 418 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi1i 447 . . . . . . . 8 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
15 elxp7 5979 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2157 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpa 291 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (2nd𝑥) ∈ 𝐵) → 𝑦𝐵)
1817adantrl 463 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) → 𝑦𝐵)
1918adantrl 463 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑦𝐵)
20 elsni 3484 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 5980 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221ancom2s 534 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((2nd𝑥) = 𝑦 ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2322an12s 533 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2420, 23sylanr2 398 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
2524adantrrr 472 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
2619, 25jca 301 . . . . . . . . . . 11 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2715, 26sylan2b 282 . . . . . . . . . 10 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2827adantl 272 . . . . . . . . 9 ((𝐴𝑉 ∧ ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
29 fveq2 5340 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30 op2ndg 5960 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
316, 30mpan2 417 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3229, 31sylan9eqr 2149 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3332adantrl 463 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
34 simprr 500 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35 snidg 3493 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3635adantr 271 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37 simprl 499 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
38 opelxpi 4499 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3936, 37, 38syl2anc 404 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
4034, 39eqeltrd 2171 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
4133, 40jca 301 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
4228, 41impbida 564 . . . . . . . 8 (𝐴𝑉 → (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4314, 42syl5bbr 193 . . . . . . 7 (𝐴𝑉 → ((𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
447, 43syl5bb 191 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4544mobidv 1991 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
465, 45mpbiri 167 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4746alrimiv 1809 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48 funcnv2 5108 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4947, 48sylibr 133 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
50 dff1o3 5294 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
513, 49, 50sylanbrc 409 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1294   = wceq 1296  wex 1433  wcel 1445  ∃*wmo 1956  Vcvv 2633  {csn 3466  cop 3469   class class class wbr 3867   × cxp 4465  ccnv 4466  cres 4469  Fun wfun 5043   Fn wfn 5044  ontowfo 5047  1-1-ontowf1o 5048  cfv 5049  1st c1st 5947  2nd c2nd 5948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950
This theorem is referenced by:  xpfi  6720  fsum2dlemstep  10977
  Copyright terms: Public domain W3C validator