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Mathbox for Alexander van der Vekens |
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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 45515 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | df-fv 6509 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
3 | 2 | eqcomi 2746 | . . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) |
4 | ifeq1 4495 | . . 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 2765 | 1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ifcif 4491 𝒫 cpw 4565 ∪ cuni 4870 class class class wbr 5110 ran crn 5639 ℩cio 6451 ‘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: dfatafv2eqfv 45567 |
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