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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 6896 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 7361 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩) | |
2 | 1 | eqeq1i 2742 | . 2 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) = 𝐶) |
3 | fnbrfvb2 6900 | . 2 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶)) | |
4 | 2, 3 | bitrid 283 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 class class class wbr 5106 × cxp 5632 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ov 7361 |
This theorem is referenced by: fnotovb 7408 |
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