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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aov0nbovbi | Structured version Visualization version GIF version |
Description: The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aov0nbovbi | ⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv0nbfvbi 43707 | . 2 ⊢ (∅ ∉ 𝐶 → ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 ↔ (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶)) | |
2 | df-aov 43677 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | 2 | eleq1i 2880 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
4 | df-ov 7138 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 4 | eleq1i 2880 | . 2 ⊢ ((𝐴𝐹𝐵) ∈ 𝐶 ↔ (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) |
6 | 1, 3, 5 | 3bitr4g 317 | 1 ⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 ∉ wnel 3091 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 '''cafv 43673 ((caov 43674 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-aiota 43642 df-dfat 43675 df-afv 43676 df-aov 43677 |
This theorem is referenced by: (None) |
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