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Theorem tz6.12c 5231
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 1946 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2 nfeu1 1927 . . . . . 6 𝑦∃!𝑦 𝐴𝐹𝑦
3 nfv 1437 . . . . . 6 𝑦 𝐴𝐹(𝐹𝐴)
42, 3nfim 1480 . . . . 5 𝑦(∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
5 tz6.12-1 5228 . . . . . . . 8 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
65expcom 113 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹𝐴) = 𝑦))
7 breq2 3796 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝑦))
87biimprd 151 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
96, 8syli 37 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
109com12 30 . . . . 5 (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
114, 10exlimi 1501 . . . 4 (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
121, 11mpcom 36 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
1312, 7syl5ibcom 148 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
1413, 6impbid 124 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wex 1397  ∃!weu 1916   class class class wbr 3792  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938
This theorem is referenced by:  fnbrfvb  5242
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