Theorem afv2fv0 47713

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 3046	. . . . . . 7 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		afv2rnfveq 47710	. . . . . . 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 2738	. . . . 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 1542   ∈ wcel 2114   ∉ wnel 3036  ∅c0 4273  ran crn 5632  ‘cfv 6498  ''''cafv2 47656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-dfat 47567  df-afv2 47657

This theorem is referenced by:  afv2fv0b  47714