Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvnufveq | Structured version Visualization version GIF version |
Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvnufveq | ⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afvfundmfveq 43331 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
2 | 1 | con3i 157 | . . 3 ⊢ (¬ (𝐹'''𝐴) = (𝐹‘𝐴) → ¬ 𝐹 defAt 𝐴) |
3 | afvnfundmuv 43332 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (¬ (𝐹'''𝐴) = (𝐹‘𝐴) → (𝐹'''𝐴) = V) |
5 | 4 | necon1ai 3043 | 1 ⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ≠ wne 3016 Vcvv 3494 ‘cfv 6349 defAt wdfat 43309 '''cafv 43310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-aiota 43279 df-dfat 43312 df-afv 43313 |
This theorem is referenced by: afvvfveq 43341 afvfv0bi 43345 aovnuoveq 43384 |
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