Theorem fnotovb 6694

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6237. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

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		opelxpi 5148	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		fnopfvb 6237	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
3	1, 2	sylan2 491	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
4	3	3impb 1260	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
5		df-ov 6653	. . 3 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
6	5	eqeq1i 2627	. 2 ⊢ ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘⟨𝐶, 𝐷⟩) = 𝑅)
7		df-ot 4186	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅⟩
8	7	eleq1i 2692	. 2 ⊢ (⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
9	4, 6, 8	3bitr4g 303	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183  ⟨cotp 4185   × cxp 5112   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)