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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfafv22 | Structured version Visualization version GIF version |
Description: Alternate definition of (𝐹''''𝐴) using (𝐹‘𝐴) directly. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
dfafv22 | ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-afv2 43407 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | df-fv 6362 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
3 | 2 | eqcomi 2830 | . . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) |
4 | ifeq1 4470 | . . 3 ⊢ ((℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
6 | 1, 5 | eqtri 2844 | 1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ifcif 4466 𝒫 cpw 4538 ∪ cuni 4837 class class class wbr 5065 ran crn 5555 ℩cio 6311 ‘cfv 6354 defAt wdfat 43314 ''''cafv2 43406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-un 3940 df-if 4467 df-fv 6362 df-afv2 43407 |
This theorem is referenced by: dfatafv2eqfv 43459 |
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