Theorem dfatafv2eqfv 47211

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.)

Assertion

Ref	Expression
dfatafv2eqfv	⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

Proof of Theorem dfatafv2eqfv

Step	Hyp	Ref	Expression
1		dfafv22 47209	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
2		iftrue 4537	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴))
3	1, 2	eqtrid 2787	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1537  ifcif 4531  𝒫 cpw 4605  ∪ cuni 4912  ran crn 5690  ‘cfv 6563   defAt wdfat 47066  ''''cafv2 47158

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-un 3968  df-if 4532  df-fv 6571  df-afv2 47159

This theorem is referenced by:  afv2rnfveq  47212  afv20fv0  47213  afv2fvn0fveq  47214