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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovpcov0 | Structured version Visualization version GIF version |
Description: If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovpcov0 | ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afvpcfv0 45626 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = V → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
2 | df-aov 45601 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | 2 | eqeq1i 2736 | . 2 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''〈𝐴, 𝐵〉) = V) |
4 | df-ov 7396 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2736 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
6 | 1, 3, 5 | 3imtr4i 291 | 1 ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 Vcvv 3473 ∅c0 4318 〈cop 4628 ‘cfv 6532 (class class class)co 7393 '''cafv 45597 ((caov 45598 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-aiota 45565 df-dfat 45599 df-afv 45600 df-aov 45601 |
This theorem is referenced by: (None) |
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