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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 7153 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | neeq1i 3080 | . 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) ≠ ∅) |
3 | afvfvn0fveq 43343 | . . 3 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → (𝐹'''〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
4 | df-aov 43314 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
5 | 3, 4, 1 | 3eqtr4g 2881 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
6 | 2, 5 | sylbi 219 | 1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 3016 ∅c0 4290 〈cop 4566 ‘cfv 6349 (class class class)co 7150 '''cafv 43310 ((caov 43311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-aiota 43279 df-dfat 43312 df-afv 43313 df-aov 43314 |
This theorem is referenced by: (None) |
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