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Theorem afv2fv0 46674
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 3053 . . . . . . 7 (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 afv2rnfveq 46671 . . . . . . 7 ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
42, 3sylbi 216 . . . . . 6 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
54eqeq1d 2730 . . . . 5 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 317 . . . 4 (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (¬ (𝐹''''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpac 477 . . 3 ((¬ (𝐹''''𝐴) = ∅ ∧ ¬ (𝐹''''𝐴) ∉ ran 𝐹) → ¬ (𝐹𝐴) = ∅)
81, 7sylbi 216 . 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 394  wo 845   = wceq 1533  wcel 2098  wnel 3043  c0 4326  ran crn 5683  cfv 6553  ''''cafv2 46617
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-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  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 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-dfat 46528  df-afv2 46618
This theorem is referenced by:  afv2fv0b  46675
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