Theorem dfatafv2eqfv 47226

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 47224	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
2		iftrue 4504	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴))
3	1, 2	eqtrid 2781	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1539  ifcif 4498  𝒫 cpw 4573  ∪ cuni 4881  ran crn 5653  ‘cfv 6528   defAt wdfat 47081  ''''cafv2 47173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3414  df-v 3459  df-un 3929  df-if 4499  df-fv 6536  df-afv2 47174

This theorem is referenced by:  afv2rnfveq  47227  afv20fv0  47228  afv2fvn0fveq  47229