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Theorem tz6.12f 5525
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 3766 . . . . 5 (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
21eleq1d 2239 . . . 4 (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3 tz6.12f.1 . . . . . . 7 𝑦𝐹
43nfel2 2325 . . . . . 6 𝑦𝐴, 𝑧⟩ ∈ 𝐹
5 nfv 1521 . . . . . 6 𝑧𝐴, 𝑦⟩ ∈ 𝐹
64, 5, 2cbveu 2043 . . . . 5 (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
76a1i 9 . . . 4 (𝑧 = 𝑦 → (∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹))
82, 7anbi12d 470 . . 3 (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)))
9 eqeq2 2180 . . 3 (𝑧 = 𝑦 → ((𝐹𝐴) = 𝑧 ↔ (𝐹𝐴) = 𝑦))
108, 9imbi12d 233 . 2 (𝑧 = 𝑦 → (((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧) ↔ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)))
11 tz6.12 5524 . 2 ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧)
1210, 11chvarv 1930 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  ∃!weu 2019  wcel 2141  wnfc 2299  cop 3586  cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206
This theorem is referenced by: (None)
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