Theorem fnotaovb 47186

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6874. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

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

Proof of Theorem fnotaovb

Step	Hyp	Ref	Expression
1		opelxpi 5656	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		fnopafvb 47143	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → ((𝐹'''⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
3	1, 2	sylan2 593	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹'''⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
4	3	3impb 1114	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹'''⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
5		df-aov 47109	. . 3 ⊢ ((𝐶𝐹𝐷)) = (𝐹'''⟨𝐶, 𝐷⟩)
6	5	eqeq1i 2734	. 2 ⊢ ( ((𝐶𝐹𝐷)) = 𝑅 ↔ (𝐹'''⟨𝐶, 𝐷⟩) = 𝑅)
7		df-ot 4586	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅⟩
8	7	eleq1i 2819	. 2 ⊢ (⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
9	4, 6, 8	3bitr4g 314	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 4583  ⟨cotp 4585   × cxp 5617   Fn wfn 6477  '''cafv 47105   ((caov 47106

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-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-aiota 47073  df-dfat 47107  df-afv 47108  df-aov 47109

This theorem is referenced by: (None)