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 6147. (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 5072 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | fnopafvb 39884 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) | |
3 | 1, 2 | sylan2 490 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
4 | 3 | 3impb 1252 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹'''〈𝐶, 𝐷〉) = 𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
5 | df-aov 39847 | . . 3 ⊢ ((𝐶𝐹𝐷)) = (𝐹'''〈𝐶, 𝐷〉) | |
6 | 5 | eqeq1i 2615 | . 2 ⊢ ( ((𝐶𝐹𝐷)) = 𝑅 ↔ (𝐹'''〈𝐶, 𝐷〉) = 𝑅) |
7 | df-ot 4134 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
8 | 7 | eleq1i 2679 | . 2 ⊢ (〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 302 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 〈cotp 4133 × cxp 5036 Fn wfn 5799 '''cafv 39843 ((caov 39844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-dfat 39845 df-afv 39846 df-aov 39847 |
This theorem is referenced by: (None) |
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