Theorem ndfatafv2undef 46594

Description: The alternate function value at a class 𝐴 is undefined if the function, whose range is a set, is not defined at 𝐴. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
ndfatafv2undef	⊢ ((ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))

Proof of Theorem ndfatafv2undef

Step	Hyp	Ref	Expression
1		ndfatafv2 46593	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		undefval 8286	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (Undef‘ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	2	eqcomd 2733	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 = (Undef‘ran 𝐹))
4	1, 3	sylan9eqr 2789	1 ⊢ ((ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  𝒫 cpw 4604  ∪ cuni 4910  ran crn 5681  ‘cfv 6551  Undefcund 8282   defAt wdfat 46498  ''''cafv2 46590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-undef 8283  df-afv2 46591

This theorem is referenced by: (None)