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Mirrors > Home > ILE Home > Th. List > fnotovb | GIF version |
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5456. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fnotovb | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4566 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | fnopfvb 5456 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → ((𝐹‘〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) | |
3 | 1, 2 | sylan2 284 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
4 | 3 | 3impb 1177 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
5 | df-ov 5770 | . . 3 ⊢ (𝐶𝐹𝐷) = (𝐹‘〈𝐶, 𝐷〉) | |
6 | 5 | eqeq1i 2145 | . 2 ⊢ ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘〈𝐶, 𝐷〉) = 𝑅) |
7 | df-ot 3532 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
8 | 7 | eleq1i 2203 | . 2 ⊢ (〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 222 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 〈cop 3525 〈cotp 3526 × cxp 4532 Fn wfn 5113 ‘cfv 5118 (class class class)co 5767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-ot 3532 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-ov 5770 |
This theorem is referenced by: (None) |
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