Theorem afv2fv0 47277

Description: If the function's value at an argument is the empty set, then the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
afv2fv0	⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))

Proof of Theorem afv2fv0

Step	Hyp	Ref	Expression
1		ioran 986	. . 3 ⊢ (¬ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹))
2		nnel 3056	. . . . . . 7 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		afv2rnfveq 47274	. . . . . . 7 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴))
4	2, 3	sylbi 217	. . . . . 6 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴))
5	4	eqeq1d 2739	. . . . 5 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅))
6	5	notbid 318	. . . 4 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (¬ (𝐹''''𝐴) = ∅ ↔ ¬ (𝐹‘𝐴) = ∅))
7	6	biimpac 478	. . 3 ⊢ ((¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹‘𝐴) = ∅)
8	1, 7	sylbi 217	. 2 ⊢ (¬ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹‘𝐴) = ∅)
9	8	con4i 114	1 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ∉ wnel 3046  ∅c0 4333  ran crn 5686  ‘cfv 6561  ''''cafv2 47220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-dfat 47131  df-afv2 47221

This theorem is referenced by:  afv2fv0b  47278