MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz6.12f Structured version   Visualization version   GIF version

Theorem tz6.12f 6931
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 4873 . . . . 5 (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
21eleq1d 2825 . . . 4 (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3 tz6.12f.1 . . . . . . 7 𝑦𝐹
43nfel2 2923 . . . . . 6 𝑦𝐴, 𝑧⟩ ∈ 𝐹
5 nfv 1913 . . . . . 6 𝑧𝐴, 𝑦⟩ ∈ 𝐹
64, 5, 2cbveuw 2605 . . . . 5 (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
76a1i 11 . . . 4 (𝑧 = 𝑦 → (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹))
82, 7anbi12d 632 . . 3 (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)))
9 eqeq2 2748 . . 3 (𝑧 = 𝑦 → ((𝐹𝐴) = 𝑧 ↔ (𝐹𝐴) = 𝑦))
108, 9imbi12d 344 . 2 (𝑧 = 𝑦 → (((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧) ↔ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)))
11 tz6.12 6930 . 2 ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧)
1210, 11chvarvv 1997 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!weu 2567  wnfc 2889  cop 4631  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator