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Theorem tz6.12-1-afv2 45721
Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6901. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
tz6.12-1-afv2 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem tz6.12-1-afv2
StepHypRef Expression
1 df-br 5142 . 2 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
21eubii 2578 . 2 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
3 tz6.12-afv2 45720 . 2 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
41, 2, 3syl2anb 598 1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ∃!weu 2561  cop 4628   class class class wbr 5141  ''''cafv2 45688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6484  df-fun 6534  df-fn 6535  df-dfat 45599  df-afv2 45689
This theorem is referenced by:  tz6.12c-afv2  45722  funressnbrafv2  45724  funbrafv2  45727
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