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Mirrors > Home > MPE Home > Th. List > fnbrfvb2 | Structured version Visualization version GIF version |
Description: Version of fnbrfvb 6711 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7194 for the form when 𝐹 is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.) |
Ref | Expression |
---|---|
fnbrfvb2 | ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5585 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑊)) | |
2 | fnbrfvb 6711 | . 2 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑊)) → ((𝐹‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) | |
3 | 1, 2 | sylan2 592 | 1 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 × cxp 5546 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: fnbrovb 7194 |
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