Theorem dfatafv2eqfv 47176

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 47174	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
2		iftrue 4554	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴))
3	1, 2	eqtrid 2792	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1537  ifcif 4548  𝒫 cpw 4622  ∪ cuni 4931  ran crn 5701  ‘cfv 6573   defAt wdfat 47031  ''''cafv2 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-un 3981  df-if 4549  df-fv 6581  df-afv2 47124

This theorem is referenced by:  afv2rnfveq  47177  afv20fv0  47178  afv2fvn0fveq  47179