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Theorem tz6.12f 5298
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 3608 . . . . 5 (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
21eleq1d 2153 . . . 4 (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3 tz6.12f.1 . . . . . . 7 𝑦𝐹
43nfel2 2237 . . . . . 6 𝑦𝐴, 𝑧⟩ ∈ 𝐹
5 nfv 1464 . . . . . 6 𝑧𝐴, 𝑦⟩ ∈ 𝐹
64, 5, 2cbveu 1969 . . . . 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 2094 . . 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 5297 . 2 ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑧)
1210, 11chvarv 1857 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  ∃!weu 1945  wnfc 2212  cop 3434  cfv 4983
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-iota 4948  df-fv 4991
This theorem is referenced by: (None)
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