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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 43765 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | df-fv 6332 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
3 | 2 | eqcomi 2807 | . . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) |
4 | ifeq1 4429 | . . 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 2821 | 1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ifcif 4425 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 ran crn 5520 ℩cio 6281 ‘cfv 6324 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-un 3886 df-if 4426 df-fv 6332 df-afv2 43765 |
This theorem is referenced by: dfatafv2eqfv 43817 |
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