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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2eqfv | Structured version Visualization version GIF version |
Description: If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2eqfv | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafv22 45565 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) | |
2 | iftrue 4497 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴)) | |
3 | 1, 2 | eqtrid 2789 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ifcif 4491 𝒫 cpw 4565 ∪ cuni 4870 ran crn 5639 ‘cfv 6501 defAt wdfat 45422 ''''cafv2 45514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-un 3920 df-if 4492 df-fv 6509 df-afv2 45515 |
This theorem is referenced by: afv2rnfveq 45568 afv20fv0 45569 afv2fvn0fveq 45570 |
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