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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfafv2 | Structured version Visualization version GIF version |
Description: Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
Ref | Expression |
---|---|
dfafv2 | ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-afv 39846 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) | |
2 | df-fv 5812 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
3 | 2 | eqcomi 2619 | . . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) |
4 | ifeq1 4040 | . . 3 ⊢ ((℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
6 | 1, 5 | eqtri 2632 | 1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ifcif 4036 class class class wbr 4583 ℩cio 5766 ‘cfv 5804 defAt wdfat 39842 '''cafv 39843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 df-fv 5812 df-afv 39846 |
This theorem is referenced by: afveq12d 39862 nfafv 39865 afvfundmfveq 39867 afvnfundmuv 39868 afvpcfv0 39875 |
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