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Mirrors > Home > MPE Home > Th. List > fnotovb | Structured version Visualization version GIF version |
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6945. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.) |
Ref | Expression |
---|---|
fnotovb | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnbrovb 7461 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷⟩𝐹𝑅)) | |
2 | df-br 5149 | . . . 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 4637 | . . . . . 6 ⊢ ⟨𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ | |
5 | 4 | eqcomi 2740 | . . . . 5 ⊢ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑅⟩ |
6 | 5 | eleq1i 2823 | . . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ⟨cop 4634 ⟨cotp 4636 class class class wbr 5148 × cxp 5674 Fn wfn 6538 (class class class)co 7412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-ov 7415 |
This theorem is referenced by: (None) |
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