Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnotaovb | Structured version Visualization version GIF version |
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6820. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
fnotaovb | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5627 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | fnopafvb 44615 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) | |
3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
4 | 3 | 3impb 1114 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
5 | df-aov 44581 | . . 3 ⊢ ((𝐶𝐹𝐷)) = (𝐹'''〈𝐶, 𝐷〉) | |
6 | 5 | eqeq1i 2745 | . 2 ⊢ ( ((𝐶𝐹𝐷)) = 𝑅 ↔ (𝐹'''〈𝐶, 𝐷〉) = 𝑅) |
7 | df-ot 4576 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
8 | 7 | eleq1i 2831 | . 2 ⊢ (〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 314 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 〈cop 4573 〈cotp 4575 × cxp 5588 Fn wfn 6427 '''cafv 44577 ((caov 44578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-br 5080 df-opab 5142 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-res 5602 df-iota 6390 df-fun 6434 df-fn 6435 df-fv 6440 df-aiota 44545 df-dfat 44579 df-afv 44580 df-aov 44581 |
This theorem is referenced by: (None) |
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