Theorem afv2ndeffv0 47852

Description: If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
afv2ndeffv0	⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅)

Proof of Theorem afv2ndeffv0

Step	Hyp	Ref	Expression
1		df-nel 3065	. . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
2		dfatafv2rnb 47819	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		df-dfat 47711	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	bitr3i 280	. . . 4 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
5	4	notbii 323	. . 3 ⊢ (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
6		ianor 997	. . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
7	1, 5, 6	3bitri 300	. 2 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
8		ndmfv 6903	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
9		nfunsn 6910	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
10	8, 9	jaoi 870	. 2 ⊢ ((¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) = ∅)
11	7, 10	sylbi 220	1 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145   ∉ wnel 3064  ∅c0 4288  {csn 4585  dom cdm 5652  ran crn 5653   ↾ cres 5654  Fun wfun 6519  ‘cfv 6525   defAt wdfat 47708  ''''cafv2 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fv 6533  df-dfat 47711  df-afv2 47801

This theorem is referenced by:  afv2fv0b  47858