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Theorem afv2fv0b 47215
Description: The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv2fv0b ((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))

Proof of Theorem afv2fv0b
StepHypRef Expression
1 afv2fv0 47214 . 2 ((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
2 afv20fv0 47212 . . 3 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
3 afv2ndeffv0 47209 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
42, 3jaoi 857 . 2 (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (𝐹𝐴) = ∅)
51, 4impbii 209 1 ((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847   = wceq 1536  wnel 3043  c0 4338  ran crn 5689  cfv 6562  ''''cafv2 47157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-dfat 47068  df-afv2 47158
This theorem is referenced by:  afv2fv0xorb  47216
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