Theorem ndfatafv2nrn 47775

Description: The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
ndfatafv2nrn	⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndfatafv2nrn

Step	Hyp	Ref	Expression
1		ndfatafv2 47765	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwuninel 8248	. . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
3		df-nel 3061	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4		eleq1 2849	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
5	4	notbid 320	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
6	3, 5	bitrid 285	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
7	2, 6	mpbiri 260	. 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
8	1, 7	syl 17	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  𝒫 cpw 4552  ∪ cuni 4862  ran crn 5644   defAt wdfat 47670  ''''cafv2 47762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nel 3061  df-rab 3414  df-v 3455  df-in 3909  df-ss 3919  df-if 4478  df-pw 4554  df-uni 4863  df-afv2 47763

This theorem is referenced by:  ndmafv2nrn  47776  nfunsnafv2  47779  dfatafv2rnb  47781  tz6.12-2-afv2  47791