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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvfundmfveq | Structured version Visualization version GIF version |
Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvfundmfveq | ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafv2 46138 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | iftrue 4533 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴)) | |
3 | 1, 2 | eqtrid 2782 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 Vcvv 3472 ifcif 4527 ‘cfv 6542 defAt wdfat 46122 '''cafv 46123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6494 df-fun 6544 df-fv 6550 df-aiota 46091 df-dfat 46125 df-afv 46126 |
This theorem is referenced by: afvnufveq 46153 afvfvn0fveq 46156 afv0nbfvbi 46157 afveu 46159 fnbrafvb 46160 afvelrn 46174 afvres 46178 tz6.12-afv 46179 dmfcoafv 46181 afvco2 46182 rlimdmafv 46183 aovfundmoveq 46187 |
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