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Theorem tz6.12 5122
 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 3735 . 2 (A𝐹y ↔ ⟨A, y 𝐹)
21eubii 1887 . 2 (∃!y A𝐹y∃!yA, y 𝐹)
3 tz6.12-1 5121 . 2 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
41, 2, 3syl2anbr 276 1 ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  ∃!weu 1878  ⟨cop 3349   class class class wbr 3734  ‘cfv 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-sn 3352  df-pr 3353  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833 This theorem is referenced by:  tz6.12f  5123
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