Theorem funopafv2b 46694

Description: Equivalence of function value and ordered pair membership, analogous to funopfvb 6948. (Contributed by AV, 6-Sep-2022.)

Assertion

Ref	Expression
funopafv2b	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Proof of Theorem funopafv2b

Step	Hyp	Ref	Expression
1		funfn 6578	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fnopafv2b 46692	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
3	1, 2	sylanb 579	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4630  dom cdm 5672  Fun wfun 6537   Fn wfn 6538  ''''cafv2 46651

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46562  df-afv2 46652

This theorem is referenced by: (None)