Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > aovvoveq | Structured version Visualization version GIF version |
Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovvoveq | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 39847 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2679 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvfveq 39877 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → (𝐹'''〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
4 | df-ov 6552 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 3, 1, 4 | 3eqtr4g 2669 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
6 | 2, 5 | sylbi 206 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 ‘cfv 5804 (class class class)co 6549 '''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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 df-fv 5812 df-ov 6552 df-afv 39846 df-aov 39847 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |