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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 44256 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = ∅ → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
2 | df-aov 44228 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | 2 | eqeq1i 2741 | . 2 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''〈𝐴, 𝐵〉) = ∅) |
4 | df-ov 7194 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2741 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
6 | 1, 3, 5 | 3imtr4i 295 | 1 ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∅c0 4223 〈cop 4533 ‘cfv 6358 (class class class)co 7191 '''cafv 44224 ((caov 44225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-aiota 44192 df-dfat 44226 df-afv 44227 df-aov 44228 |
This theorem is referenced by: (None) |
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