ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tz6.12f GIF version

Theorem tz6.12f 5278
Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
Hypothesis
Ref Expression
tz6.12f.1 𝑦𝐹
Assertion
Ref Expression
tz6.12f ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem tz6.12f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 opeq2 3597 . . . . 5 (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
21eleq1d 2151 . . . 4 (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3 tz6.12f.1 . . . . . . 7 𝑦𝐹
43nfel2 2235 . . . . . 6 𝑦𝐴, 𝑧⟩ ∈ 𝐹
5 nfv 1462 . . . . . 6 𝑧𝐴, 𝑦⟩ ∈ 𝐹
64, 5, 2cbveu 1967 . . . . 5 (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
76a1i 9 . . . 4 (𝑧 = 𝑦 → (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹))
82, 7anbi12d 457 . . 3 (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)))
9 eqeq2 2092 . . 3 (𝑧 = 𝑦 → ((𝐹𝐴) = 𝑧 ↔ (𝐹𝐴) = 𝑦))
108, 9imbi12d 232 . 2 (𝑧 = 𝑦 → (((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧) ↔ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)))
11 tz6.12 5277 . 2 ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧)
1210, 11chvarv 1855 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  ∃!weu 1943  wnfc 2210  cop 3425  cfv 4969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4934  df-fv 4977
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator