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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovnuoveq | Structured version Visualization version GIF version |
Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovnuoveq | ⊢ ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 46564 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩) | |
2 | 1 | neeq1i 2995 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ≠ V ↔ (𝐹'''⟨𝐴, 𝐵⟩) ≠ V) |
3 | afvnufveq 46590 | . . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩)) | |
4 | df-ov 7419 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩) | |
5 | 3, 1, 4 | 3eqtr4g 2790 | . 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
6 | 2, 5 | sylbi 216 | 1 ⊢ ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 2930 Vcvv 3463 ⟨cop 4630 ‘cfv 6543 (class class class)co 7416 '''cafv 46560 ((caov 46561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-aiota 46528 df-dfat 46562 df-afv 46563 df-aov 46564 |
This theorem is referenced by: (None) |
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