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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 6960. (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 7482 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷〉𝐹𝑅)) | |
| 2 | df-br 5144 | . . . 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 4635 | . . . . . 6 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
| 5 | 4 | eqcomi 2746 | . . . . 5 ⊢ 〈〈𝐶, 𝐷〉, 𝑅〉 = 〈𝐶, 𝐷, 𝑅〉 | 
| 6 | 5 | eleq1i 2832 | . . . 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 1115 | 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 〈cotp 4634 class class class wbr 5143 × cxp 5683 Fn wfn 6556 (class class class)co 7431 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: (None) | 
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