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Theorem tz6.12-1 6372
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12-1
StepHypRef Expression
1 df-fv 6057 . 2 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 iota1 6026 . . 3 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
32biimpac 504 . 2 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
41, 3syl5eq 2806 1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  ∃!weu 2607   class class class wbr 4804  cio 6010  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-sbc 3577  df-un 3720  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012  df-fv 6057
This theorem is referenced by:  tz6.12  6373  tz6.12c  6375  funbrfv  6396  setrec2lem2  42969
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