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Theorem afv2fv0b 46816
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 46815 . 2 ((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
2 afv20fv0 46813 . . 3 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
3 afv2ndeffv0 46810 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
42, 3jaoi 855 . 2 (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (𝐹𝐴) = ∅)
51, 4impbii 208 1 ((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845   = wceq 1533  wnel 3035  c0 4324  ran crn 5682  cfv 6553  ''''cafv2 46758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-dfat 46669  df-afv2 46759
This theorem is referenced by:  afv2fv0xorb  46817
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