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Theorem tz6.12f 6933
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 4879 . . . . 5 (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
21eleq1d 2824 . . . 4 (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3 tz6.12f.1 . . . . . . 7 𝑦𝐹
43nfel2 2922 . . . . . 6 𝑦𝐴, 𝑧⟩ ∈ 𝐹
5 nfv 1912 . . . . . 6 𝑧𝐴, 𝑦⟩ ∈ 𝐹
64, 5, 2cbveuw 2604 . . . . 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 2747 . . 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 6932 . 2 ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧)
1210, 11chvarvv 1996 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  wnfc 2888  cop 4637  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by: (None)
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