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Theorem tz6.12c-afv2 46460
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6904. (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 2574 . . . 4 𝑦∃!𝑦 𝐴𝐹𝑦
2 nfv 1909 . . . 4 𝑦 𝐴𝐹(𝐹''''𝐴)
3 euex 2563 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4 tz6.12-1-afv2 46459 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
54expcom 413 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6 breq2 5143 . . . . . 6 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
76biimprd 247 . . . . 5 ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
85, 7syli 39 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
91, 2, 3, 8exlimimdd 2204 . . 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 1533  ∃!weu 2554   class class class wbr 5139  ''''cafv2 46426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-res 5679  df-iota 6486  df-fun 6536  df-fn 6537  df-dfat 46337  df-afv2 46427
This theorem is referenced by:  tz6.12i-afv2  46461  dfatbrafv2b  46463  fnbrafv2b  46466  dfatcolem  46473
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