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Theorem tz6.12c-afv2 45548
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6869. (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 2587 . . . 4 𝑦∃!𝑦 𝐴𝐹𝑦
2 nfv 1918 . . . 4 𝑦 𝐴𝐹(𝐹''''𝐴)
3 euex 2576 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4 tz6.12-1-afv2 45547 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
54expcom 415 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6 breq2 5114 . . . . . 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 2213 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴))
109, 6syl5ibcom 244 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
1110, 5impbid 211 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  ∃!weu 2567   class class class wbr 5110  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-dfat 45425  df-afv2 45515
This theorem is referenced by:  tz6.12i-afv2  45549  dfatbrafv2b  45551  fnbrafv2b  45554  dfatcolem  45561
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