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Theorem tz6.12c 5515
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12c (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c
StepHypRef Expression
1 euex 2044 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2 nfeu1 2025 . . . . . 6 𝑦∃!𝑦 𝐴𝐹𝑦
3 nfv 1516 . . . . . 6 𝑦 𝐴𝐹(𝐹𝐴)
42, 3nfim 1560 . . . . 5 𝑦(∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
5 tz6.12-1 5512 . . . . . . . 8 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
65expcom 115 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹𝐴) = 𝑦))
7 breq2 3985 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝑦))
87biimprd 157 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
96, 8syli 37 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
109com12 30 . . . . 5 (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
114, 10exlimi 1582 . . . 4 (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
121, 11mpcom 36 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
1312, 7syl5ibcom 154 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
1413, 6impbid 128 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1343  wex 1480  ∃!weu 2014   class class class wbr 3981  cfv 5187
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-v 2727  df-sbc 2951  df-un 3119  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-iota 5152  df-fv 5195
This theorem is referenced by:  fnbrfvb  5526
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