Theorem tz6.12-1-afv2 47866

Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6905. (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

Step	Hyp	Ref	Expression
1		df-br 5114	. 2 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2	1	eubii 2619	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
3		tz6.12-afv2 47865	. 2 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
4	1, 2, 3	syl2anb 609	1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  ⟨cop 4600   class class class wbr 5113  ''''cafv2 47833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-dfat 47744  df-afv2 47834

This theorem is referenced by:  tz6.12c-afv2  47867  funressnbrafv2  47869  funbrafv2  47872