Theorem dfatafv2eqfv 45904

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 45902	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
2		iftrue 4533	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴))
3	1, 2	eqtrid 2785	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1542  ifcif 4527  𝒫 cpw 4601  ∪ cuni 4907  ran crn 5676  ‘cfv 6540   defAt wdfat 45759  ''''cafv2 45851

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 2704

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 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-un 3952  df-if 4528  df-fv 6548  df-afv2 45852

This theorem is referenced by:  afv2rnfveq  45905  afv20fv0  45906  afv2fvn0fveq  45907