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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovvoveq | Structured version Visualization version GIF version |
Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovvoveq | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 45665 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2824 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvfveq 45692 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → (𝐹'''〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
4 | df-ov 7397 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 3, 1, 4 | 3eqtr4g 2797 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
6 | 2, 5 | sylbi 216 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 〈cop 4629 ‘cfv 6533 (class class class)co 7394 '''cafv 45661 ((caov 45662 |
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 5293 ax-nul 5300 ax-pr 5421 |
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 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-res 5682 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7397 df-aiota 45629 df-dfat 45663 df-afv 45664 df-aov 45665 |
This theorem is referenced by: (None) |
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