Theorem tz6.12 5627

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 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12

Step	Hyp	Ref	Expression
1		df-br 4060	. 2 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2	1	eubii 2064	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
3		tz6.12-1 5626	. 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
4	1, 2, 3	syl2anbr 292	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃!weu 2055   ∈ wcel 2178  ⟨cop 3646   class class class wbr 4059  ‘cfv 5290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298

This theorem is referenced by:  tz6.12f  5628