Theorem afv2ndeffv0 43466

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 3126	. . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
2		dfatafv2rnb 43433	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		df-dfat 43325	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	bitr3i 279	. . . 4 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
5	4	notbii 322	. . 3 ⊢ (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
6		ianor 978	. . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
7	1, 5, 6	3bitri 299	. 2 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
8		ndmfv 6702	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
9		nfunsn 6709	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
10	8, 9	jaoi 853	. 2 ⊢ ((¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) = ∅)
11	7, 10	sylbi 219	1 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ∉ wnel 3125  ∅c0 4293  {csn 4569  dom cdm 5557  ran crn 5558   ↾ cres 5559  Fun wfun 6351  ‘cfv 6357   defAt wdfat 43322  ''''cafv2 43414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-dfat 43325  df-afv2 43415

This theorem is referenced by:  afv2fv0b  43472