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 6226 | . . 3 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | df-dfat 41196 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
3 | 1, 2 | sylibr 224 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
4 | afvfundmfveq 41218 | . 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 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 {csn 4177 dom cdm 5114 ↾ cres 5116 Fun wfun 5882 ‘cfv 5888 defAt wdfat 41193 '''cafv 41194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-dfat 41196 df-afv 41197 |
This theorem is referenced by: aovovn0oveq 41274 |
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