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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aov0ov0 | Structured version Visualization version GIF version |
Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aov0ov0 | ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv0fv0 45843 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = ∅ → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
2 | df-aov 45815 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | 2 | eqeq1i 2737 | . 2 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''〈𝐴, 𝐵〉) = ∅) |
4 | df-ov 7408 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2737 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
6 | 1, 3, 5 | 3imtr4i 291 | 1 ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∅c0 4321 〈cop 4633 ‘cfv 6540 (class class class)co 7405 '''cafv 45811 ((caov 45812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-aiota 45779 df-dfat 45813 df-afv 45814 df-aov 45815 |
This theorem is referenced by: (None) |
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