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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovovn0oveq | Structured version Visualization version GIF version |
Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovovn0oveq | ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6552 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | neeq1i 2846 | . 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) ≠ ∅) |
3 | afvfvn0fveq 39879 | . . 3 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → (𝐹'''〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
4 | df-aov 39847 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
5 | 3, 4, 1 | 3eqtr4g 2669 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
6 | 2, 5 | sylbi 206 | 1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ≠ wne 2780 ∅c0 3874 〈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 ax-nul 4717 ax-pow 4769 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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 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-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-fv 5812 df-ov 6552 df-dfat 39845 df-afv 39846 df-aov 39847 |
This theorem is referenced by: (None) |
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