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Theorem afv2fv0 43747
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 ((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))

Proof of Theorem afv2fv0
StepHypRef Expression
1 ioran 981 . . 3 (¬ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹))
2 nnel 3127 . . . . . . 7 (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 afv2rnfveq 43744 . . . . . . 7 ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
42, 3sylbi 220 . . . . . 6 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
54eqeq1d 2826 . . . . 5 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 321 . . . 4 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (¬ (𝐹''''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpac 482 . . 3 ((¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹𝐴) = ∅)
81, 7sylbi 220 . 2 (¬ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹𝐴) = ∅)
98con4i 114 1 ((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2115  wnel 3118  c0 4276  ran crn 5543  cfv 6343  ''''cafv2 43690
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-dfat 43601  df-afv2 43691
This theorem is referenced by:  afv2fv0b  43748
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