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Theorem tz6.12c-afv2 47254
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6928. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
tz6.12c-afv2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c-afv2
StepHypRef Expression
1 nfeu1 2588 . . . 4 𝑦∃!𝑦 𝐴𝐹𝑦
2 nfv 1914 . . . 4 𝑦 𝐴𝐹(𝐹''''𝐴)
3 euex 2577 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4 tz6.12-1-afv2 47253 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
54expcom 413 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6 breq2 5147 . . . . . 6 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
76biimprd 248 . . . . 5 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
85, 7syli 39 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
91, 2, 3, 8exlimimdd 2219 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴))
109, 6syl5ibcom 245 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
1110, 5impbid 212 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  ∃!weu 2568   class class class wbr 5143  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  tz6.12i-afv2  47255  dfatbrafv2b  47257  fnbrafv2b  47260  dfatcolem  47267
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