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Theorem tz6.12c-afv2 41922
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6399. (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 2589 . . . 4 𝑦∃!𝑦 𝐴𝐹𝑦
2 nfv 2009 . . . 4 𝑦 𝐴𝐹(𝐹''''𝐴)
3 euex 2590 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4 tz6.12-1-afv2 41921 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
54expcom 402 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6 breq2 4812 . . . . . 6 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
76biimprd 239 . . . . 5 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
85, 7syli 39 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
91, 2, 3, 8exlimimdd 2253 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴))
109, 6syl5ibcom 236 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
1110, 5impbid 203 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  ∃!weu 2580   class class class wbr 4808  ''''cafv2 41888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-res 5288  df-iota 6030  df-fun 6069  df-fn 6070  df-dfat 41799  df-afv2 41889
This theorem is referenced by:  tz6.12i-afv2  41923  dfatbrafv2b  41925  fnbrafv2b  41928  dfatcolem  41935
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