Theorem afv2fv0 47819

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 996	. . 3 ⊢ (¬ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹))
2		nnel 3070	. . . . . . 7 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		afv2rnfveq 47816	. . . . . . 7 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴))
4	2, 3	sylbi 219	. . . . . 6 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴))
5	4	eqeq1d 2763	. . . . 5 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅))
6	5	notbid 320	. . . 4 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (¬ (𝐹''''𝐴) = ∅ ↔ ¬ (𝐹‘𝐴) = ∅))
7	6	biimpac 482	. . 3 ⊢ ((¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹‘𝐴) = ∅)
8	1, 7	sylbi 219	. 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 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  ∅c0 4283  ran crn 5644  ‘cfv 6515  ''''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-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-iota 6471  df-fun 6517  df-fv 6523  df-dfat 47673  df-afv2 47763

This theorem is referenced by:  afv2fv0b  47820