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Mirrors > Home > MPE Home > Th. List > fnbrovb | Structured version Visualization version GIF version |
Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 6712 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.) |
Ref | Expression |
---|---|
fnbrovb | ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7153 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | eqeq1i 2826 | . 2 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘〈𝐴, 𝐵〉) = 𝐶) |
3 | fnbrfvb2 6716 | . 2 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) | |
4 | 2, 3 | syl5bb 285 | 1 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 × cxp 5547 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7153 |
This theorem is referenced by: fnotovb 7200 |
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