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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvfvn0fveq | Structured version Visualization version GIF version |
Description: If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvfvn0fveq | ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvfundmfvn0 6962 | . . 3 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | df-dfat 46967 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
3 | 1, 2 | sylibr 234 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
4 | afvfundmfveq 46986 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∅c0 4347 {csn 4648 dom cdm 5699 ↾ cres 5701 Fun wfun 6566 ‘cfv 6572 defAt wdfat 46964 '''cafv 46965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-res 5711 df-iota 6524 df-fun 6574 df-fv 6580 df-aiota 46933 df-dfat 46967 df-afv 46968 |
This theorem is referenced by: aovovn0oveq 47042 |
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