Theorem fnotovb 7462

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6935. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)

Assertion

Ref	Expression
fnotovb	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Proof of Theorem fnotovb

Step	Hyp	Ref	Expression
1		fnbrovb 7461	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷⟩𝐹𝑅))
2		df-br 5125	. . . 4 ⊢ (⟨𝐶, 𝐷⟩𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
3	2	a1i 11	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (⟨𝐶, 𝐷⟩𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
4		df-ot 4615	. . . . . 6 ⊢ ⟨𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅⟩
5	4	eqcomi 2745	. . . . 5 ⊢ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑅⟩
6	5	eleq1i 2826	. . . 4 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹)
7	6	a1i 11	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
8	1, 3, 7	3bitrd 305	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
9	8	3impb 1114	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4612  ⟨cotp 4614   class class class wbr 5124   × cxp 5657   Fn wfn 6531  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7413

This theorem is referenced by: (None)