Theorem dfatafv2eqfv 48152

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 48150	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
2		iftrue 4488	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴))
3	1, 2	eqtrid 2807	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ifcif 4482  𝒫 cpw 4557  ∪ cuni 4867  ran crn 5656  ‘cfv 6533   defAt wdfat 48007  ''''cafv2 48099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3904  df-if 4483  df-fv 6541  df-afv2 48100

This theorem is used by:  afv2rnfveq  48153  afv20fv0  48154  afv2fvn0fveq  48155