Theorem ndfatafv2undef 47682

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 47681	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		undefval 8223	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (Undef‘ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	2	eqcomd 2746	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 = (Undef‘ran 𝐹))
4	1, 3	sylan9eqr 2797	1 ⊢ ((ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  𝒫 cpw 4536  ∪ cuni 4845  ran crn 5626  ‘cfv 6492  Undefcund 8219   defAt wdfat 47586  ''''cafv2 47678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-undef 8220  df-afv2 47679

This theorem is referenced by: (None)