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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 43483 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | df-fv 6356 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
3 | 2 | eqcomi 2829 | . . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) |
4 | ifeq1 4464 | . . 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 2843 | 1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ifcif 4460 𝒫 cpw 4532 ∪ cuni 4831 class class class wbr 5059 ran crn 5549 ℩cio 6305 ‘cfv 6348 defAt wdfat 43390 ''''cafv2 43482 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-un 3934 df-if 4461 df-fv 6356 df-afv2 43483 |
This theorem is referenced by: dfatafv2eqfv 43535 |
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