Theorem tz6.12 5297

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 3823	. 2 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2	1	eubii 1954	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
3		tz6.12-1 5296	. 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
4	1, 2, 3	syl2anbr 286	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1287   ∈ wcel 1436  ∃!weu 1945  ⟨cop 3434   class class class wbr 3822  ‘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-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-uni 3639  df-br 3823  df-iota 4948  df-fv 4991

This theorem is referenced by:  tz6.12f  5298