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Theorem tz6.12c 6251
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 2522 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2 nfeu1 2508 . . . . . 6 𝑦∃!𝑦 𝐴𝐹𝑦
3 nfv 1883 . . . . . 6 𝑦 𝐴𝐹(𝐹𝐴)
42, 3nfim 1865 . . . . 5 𝑦(∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
5 tz6.12-1 6248 . . . . . . . 8 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
65expcom 450 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹𝐴) = 𝑦))
7 breq2 4689 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝑦))
87biimprd 238 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
96, 8syli 39 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
109com12 32 . . . . 5 (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
114, 10exlimi 2124 . . . 4 (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
121, 11mpcom 38 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
1312, 7syl5ibcom 235 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
1413, 6impbid 202 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wex 1744  ∃!weu 2498   class class class wbr 4685  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by:  tz6.12i  6252  fnbrfvb  6274
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