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Theorem tz6.12 6354
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
Assertion
Ref Expression
tz6.12 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12
StepHypRef Expression
1 df-br 4788 . 2 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
21eubii 2640 . 2 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
3 tz6.12-1 6353 . 2 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
41, 2, 3syl2anbr 586 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  ∃!weu 2618  cop 4323   class class class wbr 4787  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-un 3728  df-sn 4318  df-pr 4320  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038
This theorem is referenced by:  tz6.12f  6355  dfac5lem5  9154  tz6.12-afv  41770
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