![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > aovfundmoveq | Structured version Visualization version GIF version |
Description: If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovfundmoveq | ⊢ (𝐹 defAt 〈𝐴, 𝐵〉 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afvfundmfveq 42873 | . 2 ⊢ (𝐹 defAt 〈𝐴, 𝐵〉 → (𝐹'''〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
2 | df-aov 42856 | . 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | df-ov 7019 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
4 | 1, 2, 3 | 3eqtr4g 2856 | 1 ⊢ (𝐹 defAt 〈𝐴, 𝐵〉 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 〈cop 4478 ‘cfv 6225 (class class class)co 7016 defAt wdfat 42851 '''cafv 42852 ((caov 42853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-int 4783 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-res 5455 df-iota 6189 df-fun 6227 df-fv 6233 df-ov 7019 df-aiota 42821 df-dfat 42854 df-afv 42855 df-aov 42856 |
This theorem is referenced by: aovmpt4g 42936 |
Copyright terms: Public domain | W3C validator |