Theorem tz6.12-1 5441

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) → (𝐹‘𝐴) = 𝑦)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12-1

Step	Hyp	Ref	Expression
1		df-fv 5126	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		iota1 5097	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
3	2	biimpac 296	. 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
4	1, 3	syl5eq 2182	1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃!weu 1997   class class class wbr 3924  ℩cio 5081  ‘cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083  df-fv 5126

This theorem is referenced by:  tz6.12  5442  tz6.12c  5444  funbrfv  5453