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Theorem tz6.12c 6436
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 nfeu1 2649 . . . 4 𝑦∃!𝑦 𝐴𝐹𝑦
2 nfv 2005 . . . 4 𝑦 𝐴𝐹(𝐹𝐴)
3 euex 2663 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4 tz6.12-1 6433 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
54expcom 400 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹𝐴) = 𝑦))
6 breq2 4855 . . . . . 6 ((𝐹𝐴) = 𝑦 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝑦))
76biimprd 239 . . . . 5 ((𝐹𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
85, 7syli 39 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
91, 2, 3, 8exlimimdd 2257 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
109, 6syl5ibcom 236 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
1110, 5impbid 203 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  ∃!weu 2637   class class class wbr 4851  cfv 6104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-iota 6067  df-fv 6112
This theorem is referenced by:  tz6.12i  6437  fnbrfvb  6459
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