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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 5471. (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 4579 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | fnopfvb 5471 | . . . 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 1178 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
5 | df-ov 5785 | . . 3 ⊢ (𝐶𝐹𝐷) = (𝐹‘〈𝐶, 𝐷〉) | |
6 | 5 | eqeq1i 2148 | . 2 ⊢ ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘〈𝐶, 𝐷〉) = 𝑅) |
7 | df-ot 3542 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
8 | 7 | eleq1i 2206 | . 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 963 = wceq 1332 ∈ wcel 1481 〈cop 3535 〈cotp 3536 × cxp 4545 Fn wfn 5126 ‘cfv 5131 (class class class)co 5782 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-ot 3542 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-ov 5785 |
This theorem is referenced by: (None) |
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